fa نابخش‌پذیری گروه‌های آبلی جبر alg علمی پژوهشی بنیادی S <span style="font-size:11pt"><span style="line-height:21.0pt"><span style="tab-stops:right 26.05pt 396.85pt 437.1pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><span style="font-family:Calibri,sans-serif"><span lang="AR-SA" style="font-size:10.0pt"><span b="" nazanin="" style="font-family:">در سرتاسر متن گروه‌ها آبلی هستند.‌ گروه </span></span><span dir="LTR" style="font-size:10.0pt">G</span><span lang="AR-SA" style="font-size:10.0pt"><span b="" nazanin="" style="font-family:"> را </span></span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&ndash;</span></span><span dir="LTR" style="font-size:10.0pt">n</span><span lang="AR-SA" style="font-size:10.0pt"><span b="" nazanin="" style="font-family:"> بخش‌پذیر گوئیم هرگاه.‌ گروه </span></span><span dir="LTR" style="font-size:10.0pt">G</span><span lang="AR-SA" style="font-size:10.0pt"><span b="" nazanin="" style="font-family:"> را مطلقاً نابخش‌پذیر گوئیم هرگاه برای هر، فاقد زیرگروه ناصفر </span></span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&ndash;</span></span><span dir="LTR" style="font-size:10.0pt">n</span><span lang="AR-SA" style="font-size:10.0pt"><span b="" nazanin="" style="font-family:"> بخش‌پذیر باشد.‌ در بررسی کلاس </span></span><span dir="LTR" style="font-size:10.0pt">C</span><span lang="AR-SA" style="font-size:10.0pt"><span b="" nazanin="" style="font-family:"> متشکل از تمام گروه‌های مطلقاً نابخش‌پذیر مانند </span></span><span dir="LTR" style="font-size:10.0pt">G</span><span lang="AR-SA" style="font-size:10.0pt"><span b="" nazanin="" style="font-family:">، به زیرگروه‌های جمع تمام زیرگروه‌های -</span></span><span dir="LTR" style="font-size:10.0pt">p</span><span lang="AR-SA" style="font-size:10.0pt"><span b="" nazanin="" style="font-family:"> بخش‌پذیر و (برای هر عدد اوّل </span></span><span dir="LTR" style="font-size:10.0pt">p</span><span lang="AR-SA" style="font-size:10.0pt"><span b="" nazanin="" style="font-family:">) بر می‌خوریم.‌ خواص این دو زیرگروه به تفصیل مورد بررسی قرار گرفته است و برای کلاس تمام گروه‌های&nbsp; بخش‌پذیر&nbsp; و کلاس متشکل از تمام گروه‌ها با، ثابت می‌کنیم زوج یک نظریه تاب است.‌ کلاس </span></span><span dir="LTR" style="font-size:10.0pt">C</span><span lang="AR-SA" style="font-size:10.0pt"><span b="" nazanin="" style="font-family:"> تحت هر جمع مستقیم و هر حاصل‌ضرب بسته است و اگر آن‌گاه نشان می‌دهیم.‌ همچنین ثابت می‌شود که اگر و تنها اگر برای هر </span></span><span dir="LTR" style="font-size:10.0pt">p</span><span lang="AR-SA" style="font-size:10.0pt"><span b="" nazanin="" style="font-family:">، اگر و تنها اگر.‌ سرانجام مشخص‌سازی دیگری برای زیرگروه‌هایی از </span></span><span dir="LTR" style="font-size:10.0pt">Q</span><span lang="AR-SA" style="font-size:10.0pt"><span b="" nazanin="" style="font-family:"> (اعداد گویا) که به </span></span><span dir="LTR" style="font-size:10.0pt">C</span><span lang="AR-SA" style="font-size:10.0pt"><span b="" nazanin="" style="font-family:"> تعلق دارند، بیان شده است.‌ مثال‌های متنوع نیز جهت توصیف نتایج آورده شده است.‌</span></span></span></span></span></span></span></span><br> <br> &nbsp; <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">Introduction</span></span></b></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">&nbsp; In Throughout all groups are abelian. Suppose that G is a group and n is a positive integer. For a </span></span><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:">&isin;</span></span><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> G, if we consider the solution of the equation nx = a in G, two subsets of G are proposed. One of them is {a </span></span><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:">&isin;</span></span><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> G | </span></span><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:">&exist;</span></span><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">x </span></span><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:">&isin;</span></span><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> G,nx = a} and the other is {x </span></span><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:">&isin;</span></span><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> G | nx = a} for given a </span></span><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:">&isin;</span></span><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> G. The first is nG, which is clearly a subgroup of G, but the second does not have to be a subgroup. However, if we replace the equation nx = a with nx </span></span><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:">&isin;</span></span><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">< a > then we come to the equation nx = 0 in the group, whose solutions determine a subgroup of&nbsp;&nbsp; (hence of G). In this regard, we state something about divisibility from [2]. Let a is an element in a group G. The element a is called divisible whenever for every n &ge; 1 there exists x </span></span><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:">&isin;</span></span><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> G such that nx = a. Also a is called torsion whenever there exists positive integer m such that a is a solution of the equation mx = 0. The group G is then called divisible (resp. torsion) if every element in G is divisible (resp. torsion). Furthermore, G is called reduced (resp. torsionfree) if it has no non-zero divisible (resp. torsion) subgroup. <span style="letter-spacing:-.2pt">Therefore, G is divisible if and only if nG = G for every n &ge; 1. As canonical examples, we can mention the additive group Q and. Here, is the subgroup of Q/Z generated by {1/pi + }. Also, and all proper subgroup of Q are reduced; [1] and [2] are excellent references on the subject.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">Suppose that n &ge; 1. It is easy to verify that nG = G if and only if pG = G for every prime number p | n. This follows that G is divisible if and only if pG = G for every prime number p. Thus G is non &minus; divisible if there exists a prime number q such that qG &ne; G. Based on the above, we may define the divisibility (non-divisibility) with respect to a number.</span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">Definition 1.1. Let n &ge; 1 a group G is called:</span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">(a)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; n-divisible if nG = G.</span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">(b)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Fully non-divisible if pG &ne; G for every prime number p.</span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">(c)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Absolutely non-divisible if pH &ne; H for every prime number p and non-zero subgroup H of G.</span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">Thus, we deal with three class of groups as blow:</span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times="">{Absolutely non-divisible groups} </span></span><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:">&sube;</span></span><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> {Fully non-divisible} </span></span><span style="font-size:9.0pt">&cap;</span><span style="font-size:9.0pt"><span new="" roman="" style="font-family:" times=""> {Reduced groups}. Examples are presented to show that these three classes are mutually distinct.</span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:9.0pt"><span cambria="" style="font-family:">main results</span></span></b><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:9.0pt"><span cambria="" style="font-family:">Definition 2.1. </span></span></b><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">For every prime number p, let rad_p</span></span></i><span style="font-size:9.0pt"><span cambria="" style="font-family:">(<i>G</i>) = &cap;<i>_n</i>_&ge;1<i>pⁿG and T_p</i>(<i>G</i>)<i>, the sum of all p-divisible subgroups of G. D_nbe the class of all n-divisible groups and F_pbe the class of all groups G with T_p</i>(<i>G</i>) = {0}<i>. Let </i></span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r><m:r>={</m:r><m:r>G</m:r><m:r>|</m:r><m:r>G</m:r><m:r>=</m:r></span></span></i><m:nary><m:narypr><m:chr m:val="∑"><m:suphide m:val="on"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:suphide></m:chr></m:narypr><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>H</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r>⩽</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:sub><m:sup></m:sup><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>H</m:r></span></span></i></m:e></m:nary></m:omath><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">&nbsp;such that H </span></span></i><span style="font-size:9.0pt"><span cambria="" style="font-family:">&isin; &cup;<i>_n</i>_&ge;1<i>D_n</i>} <i>and C_pbe the class of all groups G with rad_p</i>(<i>G</i>) = {0}<i>.</i></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt">T</span><b><span style="font-size:9.0pt"><span cambria="" style="font-family:">heorem 2.2. </span></span></b><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">Let p be a prime number.</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span> <ol> <li style="list-style-type:none"> <ol style="list-style-type:lower-alpha"> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span cambria="" style="font-family:"><span style="color:black"></span></span></span><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><span style="letter-spacing:-.2pt">For every group homomorphism f</span></span></span></i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><span style="letter-spacing:-.2pt">: <i>G</i>₁&rarr; <i>G</i>₂<i>we have f </i>(rad<i>_p</i>(<i>G</i>₁)) &sube; rad<i>_p</i>(<i>G</i>₂)<i>. Furthermore rad_p</i>(<i>G</i>) <i>is a fully invariant subgroup of G.</i></span></span></span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">For every </span></span></i><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>H</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r>⩽</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">&nbsp;we have </span></span></i><span style="font-size:9.0pt"><span cambria="" style="font-family:">rad<i>_p</i>(<i>H</i>) &sube; rad<i>_p</i>(<i>G</i>)<i>. Also if G </i>= <i>H</i>&oplus;<i>K then </i>rad<i>_p</i>(<i>H</i>) = <i>H </i>&cap; rad<i>_p</i>(<i>G</i>)<i>.</i></span></span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><m:omath><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>ra</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>d</m:r></span></span></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r>&oplus;</m:r></span></span></i></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r>&isin;</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>I</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>)=</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r>&oplus;</m:r></span></span></i></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r>&isin;</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>I</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>ra</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>d</m:r></span></span></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>).</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><m:omath><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>ra</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>d</m:r></span></span></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r></span></span></i><m:nary><m:narypr><m:chr m:val="∏"><m:suphide m:val="on"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:suphide></m:chr></m:narypr><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r>&isin;</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>I</m:r></span></span></i></m:sub><m:sup></m:sup><m:e><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sub></m:ssub><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:e></m:nary><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>)=</m:r></span></span></i><m:nary><m:narypr><m:chr m:val="∏"><m:suphide m:val="on"><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:suphide></m:chr></m:narypr><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r>&isin;</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>I</m:r></span></span></i></m:sub><m:sup></m:sup><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>ra</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>d</m:r></span></span></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:e></m:nary><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>)</m:r></span></span></i></m:omath><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">.</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">pG </span></span></i><span style="font-size:9.0pt"><span cambria="" style="font-family:">= <i>G if and only if </i>rad<i>_p</i>(<i>G</i>) = <i>G if and only if </i>&nbsp;</span></span><m:omath><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>Ho</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>m</m:r></span></span></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>Z</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>(</m:r><m:r>G</m:r><m:r>,</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>Z</m:r></span></span></i></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>)={0}</m:r></span></span></i></m:omath><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">.</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">For every </span></span></i><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>H</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r>⩽</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">&nbsp;we have</span></span></i><m:omath><m:f><m:fpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:fpr><m:num><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>ra</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>d</m:r></span></span></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r><m:r>G</m:r><m:r>)+</m:r><m:r>H</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:num><m:den><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>H</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:den></m:f><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r>&sube;</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>ra</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>d</m:r></span></span></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r></span></span></i><m:f><m:fpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:fpr><m:num><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:num><m:den><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>H</m:r></span></span></i></m:den></m:f><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>)</m:r></span></span></i></m:omath><span dir="RTL" lang="FA" style="font-size:9.0pt"><span cambria="" style="font-family:">.</span></span><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"> Also, if </span></span></i><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>H</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r>&sube;</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>ra</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>d</m:r></span></span></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></span></span></i></m:omath><span style="font-size:9.0pt"><span cambria="" style="font-family:">&nbsp;<i>then</i></span></span><m:omath><m:f><m:fpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:fpr><m:num><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>ra</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>d</m:r></span></span></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:num><m:den><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>H</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:den></m:f><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>=</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>ra</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>d</m:r></span></span></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r></span></span></i><m:f><m:fpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:fpr><m:num><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:num><m:den><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>H</m:r></span></span></i></m:den></m:f><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>)</m:r></span></span></i></m:omath><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">. Furthermore</span></span></i><m:omath><m:f><m:fpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:fpr><m:num><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:num><m:den><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>ra</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>d</m:r></span></span></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></span></span></i></m:den></m:f><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>={0}</m:r></span></span></i></m:omath><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">&nbsp;.</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><m:omath><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>ra</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>d</m:r></span></span></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r><m:r>G</m:r><m:r>)=</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>Rej</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>C</m:r></span></span></i></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>)</m:r></span></span></i></m:omath><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">.</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><m:omath><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>ra</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>d</m:r></span></span></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r><m:r>G</m:r><m:r>)=</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>Rej</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,{</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Z</m:r></span></span></m:e><m:sub><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i></m:e><m:sup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sup></m:ssup><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>}</m:r></span></span></i></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r>&ge;</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>1</m:r></span></span></i></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>)</m:r></span></span></i></m:omath><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">.</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span></li> </ol> </li> </ol> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:9.0pt"><span cambria="" style="font-family:">Theorem 2.3. </span></span></b><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">Let p be a prime number.</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span> <ol> <li style="list-style-type:none"> <ol style="list-style-type:lower-alpha"> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">The class of p-divisible groups is closed under direct sum and homomorphic image.</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">For every group G, </span></span></i><span style="font-size:9.0pt"><span cambria="" style="font-family:">T<i>_p</i>(<i>G</i>) <i>is p-divisible and we have </i>T<i>_p</i>(<i>G</i>) &sube; rad<i>_p</i>(<i>G</i>)<i>. Furthermore </i>T<i>_p</i>(rad<i>_p</i>(<i>G</i>)) = rad<i>_p</i>(T<i>_p</i>(<i>G</i>)) = T<i>_p</i>(<i>G</i>)<i>.</i></span></span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">If G is a p-torsionfree group, then </span></span></i><span style="font-size:9.0pt"><span cambria="" style="font-family:">rad<i>_p</i>(<i>G</i>) <i>is a p-divisible subgroup and </i>rad<i>_p</i>(<i>G</i>) = T<i>_p</i>(<i>G</i>)<i>.</i></span></span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">Let G be a p-torsionfree group and H </span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:">&le;</span></span> <i><span style="font-size:9.0pt"><span cambria="" style="font-family:">G. H </span></span></i><span style="font-size:9.0pt"><span cambria="" style="font-family:">&sube; rad<i>_p</i>(<i>G</i>) <i>if and only if </i>rad<i>_p</i>(<i>G</i>) = rad<i>_p</i>(<i>H</i>)<i>. Furthermore </i>rad<i>_p</i>(rad<i>_p</i>(<i>G</i>)) = rad<i>_p</i>(<i>G</i>)<i>.</i></span></span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">If&nbsp; </span></span></i><span style="font-size:9.0pt"><span cambria="" style="font-family:">T<i>_p</i>(<i>G</i>) = {0}<i>, then p divide the order of every torsion element in G.</i></span></span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">Let p and q be two different prime numbers. If </span></span></i><span style="font-size:9.0pt"><span cambria="" style="font-family:">T<i>_p</i>(<i>G</i>) = T<i>_q</i>(<i>G</i>) = {0}<i>, then </i>rad<i>_p</i>(<i>G</i>) = rad<i>_q</i>(<i>G</i>) = {0}<i>.</i></span></span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>T</m:r></span></span></i></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r></span></span></i><m:f><m:fpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:fpr><m:num><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:num><m:den><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>T</m:r></span></span></i></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></span></span></i></m:den></m:f><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>)={0}</m:r></span></span></i></m:omath><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">.</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span></li> </ol> </li> </ol> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:9.0pt"><span cambria="" style="font-family:">Theorem 2.4. </span></span></b><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">For every prime number p, </span></span></i><span style="font-size:9.0pt"><span cambria="" style="font-family:">(<i>D_p,F_p</i>) <i>is a torsion theory.</i></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:9.0pt"><span cambria="" style="font-family:">Theorem 2.5. </span></span></b><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">Every absolutely non-divisible group G is torsion free and so G is isomorphic to a subgroup of</span></span></i><m:omath><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>Q</m:r></span></span></i></m:e><m:sup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>&Lambda;</m:r></span></span></i></m:sup></m:ssup></m:omath><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">.</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:9.0pt"><span cambria="" style="font-family:">Theorem 2.6. </span></span></b><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">The following statements are equivalent for every group G.</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span> <ol> <li style="list-style-type:none"> <ol style="list-style-type:lower-alpha"> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">G is absolutely non-divisible,</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">for every prime number p, </span></span></i><span style="font-size:9.0pt"><span cambria="" style="font-family:">rad<i>_q</i>(<i>G</i>) = {0}<i>,</i></span></span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span><m:omath><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>Ho</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:nor></m:nor></m:rpr>m</m:r></span></span></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>Z</m:r></span></span></i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>(</m:r><m:r>D</m:r><m:r>,</m:r><m:r>G</m:r><m:r>)={0}</m:r></span></span></i></m:omath><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">.</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span></li> </ol> </li> </ol> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:9.0pt"><span cambria="" style="font-family:">Theorem 2.7. </span></span></b><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">The class of absolutely non-divisible is closed under direct product and subgroup.</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt">T</span><b><span style="font-size:9.0pt"><span cambria="" style="font-family:">heorem 2.8. </span></span></b><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">If H and G/H are absolutely non-divisible groups then G is absolutely non-divisible group.</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:9.0pt"><span cambria="" style="font-family:">Theorem 2.9. </span></span></b><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">For every group G the following statements hold.</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">.</span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria",serif"></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span cambria="" style="font-family:">(b) <i>G is an absolutely non-divisible group if and only if for every prime number p there exists a natural number n such that </i><i>&nbsp;is absolutely non-divisible.</i></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:9.0pt"><span cambria="" style="font-family:">For </span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>H</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r>⩽</m:r><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>Q</m:r></span></span></i></m:omath>&nbsp;<span style="font-size:9.0pt"><span cambria="" style="font-family:">and prime number <i>p</i>, let</span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>B</m:r></span></span></i></m:e><m:sub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i></m:sub></m:ssub><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>(</m:r><m:r>H</m:r><m:r>)={</m:r><m:r>t</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>&isin;N</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>|</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>&exist;</m:r></span></span></i><m:f><m:fpr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:fpr><m:num><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>m</m:r></span></span></i></m:num><m:den><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:den></m:f><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r>&isin;</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>H</m:r><m:r>,(</m:r><m:r>m</m:r><m:r>,</m:r><m:r>n</m:r><m:r>)=1,</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></i></m:e><m:sup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>t</m:r></span></span></i></m:sup></m:ssup><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r>|</m:r><m:r>n</m:r><m:r>}</m:r></span></span></i></m:omath><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">, </span></span></i><span style="font-size:9.0pt"><span cambria="" style="font-family:">and <i>b_p</i>(<i>H</i>) = |<i>B_p</i>(<i>H</i>)|.</span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:18.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:9.0pt"><span cambria="" style="font-family:">Theorem 2.10. </span></span></b><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">Let </span></span></i><span style="font-size:9.0pt"><span cambria="" style="font-family:">{0} &ne; <i>G &le; </i></span></span><m:omath><i><span style="font-size:9.0pt"><span cambria="" style="font-family:"><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>Q</m:r></span></span></i><i><span style="font-size:9.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr> </m:r></span></span></i></m:omath><i><span style="font-size:9.0pt"><span cambria="" style="font-family:">G is absolutely non-divisible if and only if for every prime number p, b_p</span></span></i><span style="font-size:9.0pt"><span cambria="" style="font-family:">(<i>G</i>)<i><</i>&infin;<i>.</i></span></span></span></span></span><br> <br> &nbsp; گروه بخش‌پذیر, رادیکال, مطلقاً نابخش‌پذیر, تماماً نابخش‌پذیر. Absolutely non-divisible group, fully non-divisible group, p-divisible group, pradical. 266 280 http://mmr.khu.ac.ir/browse.php?a_code=A-10-1271-1&slc_lang=fa&sid=1 mohammad reza vedadi محمد رضا ودادی mrvedadi@iut.ac.ir 10031947532846005485 10031947532846005485 Yes Isfahan University of Technology دانشگاه صنعتی اصفهان yaser Tolooei یاسر طلوعی y.toloei@razi.ac.ir 10031947532846005486 10031947532846005486 No Razi University دانشگاه رازی