Mathematical Researches

en مشخصه ی جدیدی از بعضی ازگروه های ساده متناهی A new recognition of some finite simple groups ریاضی Mat مقاله استخراج شده از طرح پژوهشی Research Paper بعد از رده بندی گروه های ساده متناهی، یکی از مسایل مهم که مورد بحث محققین قرارگرفت مساله  تشخیص پذیری یک گروه با یک ویژگی خاص است، در واقع گروه <img alt="" chromakey="white" src="file:///C:Users10AppDataLocalTempmsohtmlclip11clip_image001.png" >  با خاصیت <img alt="" chromakey="white" src="file:///C:Users10AppDataLocalTempmsohtmlclip11clip_image002.png" > تشخیص پذیر است هرگاه گروه <img alt="" chromakey="white" src="file:///C:Users10AppDataLocalTempmsohtmlclip11clip_image001.png" >  تحت یکریختی تنها گروهی با خاصیت <img alt="" chromakey="white" src="file:///C:Users10AppDataLocalTempmsohtmlclip11clip_image003.png" >  باشد. تا کنون در این زمینه تشخیص پذیرهای زیادی با روشهای مختلف توسط محققین مورد بررسی قرار گرفته شده است؛ به عنوان مثال تشخیص پذیری با استفاده از مرتبه ی عناصر، تعداد عناصر هم مرتبه، بزرگترین مرتبه ی عناصر، انواع مختلف گرافهاو... مورد بررسی قرار گرفته شده است. دراین مقاله ثابت می کنیم که گروه های استینبرگ ( <img alt="" chromakey="white" src="file:///C:Users10AppDataLocalTempmsohtmlclip11clip_image004.png" > که درآن <img alt="" chromakey="white" src="file:///C:Users10AppDataLocalTempmsohtmlclip11clip_image005.png" >   عددی اول است، با استفاده از مرتبه ی گروه و بزرگترین مرتبه ی عناصرگروه تشخیص پذیر است.بعد از رده بندی گروه های ساده متناهی، یکی از مسایل مهم که مورد بحث محققین قرارگرفت مساله  تشخیص پذیری یک گروه با یک ویژگی خاص است، در واقع گروه <img alt="" chromakey="white" src="file:///C:Users10AppDataLocalTempmsohtmlclip11clip_image001.png" >  با خاصیت <img alt="" chromakey="white" src="file:///C:Users10AppDataLocalTempmsohtmlclip11clip_image002.png" > تشخیص پذیر است هرگاه گروه <img alt="" chromakey="white" src="file:///C:Users10AppDataLocalTempmsohtmlclip11clip_image001.png" >  تحت یکریختی تنها گروهی با خاصیت <img alt="" chromakey="white" src="file:///C:Users10AppDataLocalTempmsohtmlclip11clip_image003.png" >  باشد. تا کنون در این زمینه تشخیص پذیرهای زیادی با روشهای مختلف توسط محققین مورد بررسی قرار گرفته شده است؛ به عنوان مثال تشخیص پذیری با استفاده از مرتبه ی عناصر، تعداد عناصر هم مرتبه، بزرگترین مرتبه ی عناصر، انواع مختلف گرافهاو... مورد بررسی قرار گرفته شده است. دراین مقاله ثابت می کنیم که گروه های استینبرگ ( <img alt="" chromakey="white" src="file:///C:Users10AppDataLocalTempmsohtmlclip11clip_image004.png" > که درآن <img alt="" chromakey="white" src="file:///C:Users10AppDataLocalTempmsohtmlclip11clip_image005.png" >   عددی اول است، با استفاده از مرتبه ی گروه و بزرگترین مرتبه ی عناصرگروه تشخیص پذیر است. Introduction In this paper G is considered to be a finite group. We denote the set of elements order and the set of prime divisors of order G by <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>π</m:r></m:e><m:sub><m:r>e</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d><m:r> </m:r><m:r>and</m:r><m:r> </m:r><m:r>π</m:r><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></m:omath> , respectively. The largest element order of G is denoted by <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>k</m:r></m:e><m:sub><m:r>1</m:r></m:sub></m:ssub></m:omath> (G), and the prime graph of G is denoted by <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Γ</m:r><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></m:omath> , where two vertices u and v are adjacent if uv<m:omath><m:r> ∈</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>π</m:r></m:e><m:sub><m:r>e</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d></m:omath> . After classification of finite simple groups, the problem that came to the researchers’ attention was the problem of recognizing a group with a specific characteristic. Properties such as elements order, the set of elements with the same order, graphs, etc. In fact we say the group G by property M is recognizable, whenever by isomorphic G be the only group by property M. The other methods are group recognition by using the order of the group and the largest element order. In other words, we say the group G is recognizably by using the order of the G and the largest element order whenever there exists group H so that<m:omath><m:r> ∣</m:r><m:r>G</m:r><m:r>∣=</m:r></m:omath> ∣H∣ and <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>k</m:r></m:e><m:sub><m:r>1</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d><m:r>=</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>k</m:r></m:e><m:sub><m:r>1</m:r></m:sub></m:ssub></m:omath> (H) then  <m:omath><m:r>G</m:r><m:r>&cong;</m:r></m:omath>   H. It is known that some of groups are recognizable by this method. In this paper, we prove that the Stienberg group<m:omath><m:spre><m:sprepr><m:ctrlpr></m:ctrlpr></m:sprepr><m:sub><m:r> </m:r></m:sub><m:sup><m:r>3</m:r></m:sup><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>D</m:r></m:e><m:sub><m:r>4</m:r></m:sub></m:ssub></m:e></m:spre><m:r>(</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>2</m:r></m:e><m:sup><m:r>n</m:r></m:sup></m:ssup></m:omath> ), where <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>2</m:r></m:e><m:sup><m:r>4</m:r><m:r>n</m:r></m:sup></m:ssup><m:r>-</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>2</m:r></m:e><m:sup><m:r>2</m:r><m:r>n</m:r></m:sup></m:ssup><m:r>+1</m:r></m:omath>  is a prime number are recognizable by using the order of the group and the largest element order. In other words, we have the following main theorem. Main Theorem  Let G be a group with the Steinberg group <m:omath><m:spre><m:sprepr><m:ctrlpr></m:ctrlpr></m:sprepr><m:sub><m:r> </m:r></m:sub><m:sup><m:r>3</m:r></m:sup><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>D</m:r></m:e><m:sub><m:r>4</m:r></m:sub></m:ssub></m:e></m:spre><m:r>(</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>2</m:r></m:e><m:sup><m:r>n</m:r></m:sup></m:ssup></m:omath> ), where<m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>2</m:r></m:e><m:sup><m:r>4</m:r><m:r>n</m:r></m:sup></m:ssup><m:r>-</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>2</m:r></m:e><m:sup><m:r>2</m:r><m:r>n</m:r></m:sup></m:ssup><m:r>+1</m:r></m:omath>  is a prime number such that <m:omath><m:r>∣</m:r><m:r>G</m:r><m:r>∣=</m:r></m:omath> ∣<m:omath><m:spre><m:sprepr><m:ctrlpr></m:ctrlpr></m:sprepr><m:sub><m:r> </m:r></m:sub><m:sup><m:r>3</m:r></m:sup><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>D</m:r></m:e><m:sub><m:r>4</m:r></m:sub></m:ssub></m:e></m:spre><m:r>(</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>2</m:r></m:e><m:sup><m:r>n</m:r></m:sup></m:ssup></m:omath> )∣ and <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>k</m:r></m:e><m:sub><m:r>1</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d><m:r>=</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>k</m:r></m:e><m:sub><m:r>1</m:r></m:sub></m:ssub></m:omath> (<m:omath><m:spre><m:sprepr><m:ctrlpr></m:ctrlpr></m:sprepr><m:sub><m:r> </m:r></m:sub><m:sup><m:r>3</m:r></m:sup><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>D</m:r></m:e><m:sub><m:r>4</m:r></m:sub></m:ssub></m:e></m:spre><m:r>(</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>2</m:r></m:e><m:sup><m:r>n</m:r></m:sup></m:ssup></m:omath> )), then <m:omath><m:r>G</m:r><m:r>&cong;</m:r></m:omath>   <m:omath><m:spre><m:sprepr><m:ctrlpr></m:ctrlpr></m:sprepr><m:sub><m:r> </m:r></m:sub><m:sup><m:r>3</m:r></m:sup><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>D</m:r></m:e><m:sub><m:r>4</m:r></m:sub></m:ssub></m:e></m:spre><m:r>(</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>2</m:r></m:e><m:sup><m:r>n</m:r></m:sup></m:ssup></m:omath> ). Material and methods In this research we prove that Steinberg group<m:omath><m:spre><m:sprepr><m:ctrlpr></m:ctrlpr></m:sprepr><m:sub><m:r> </m:r></m:sub><m:sup><m:r>3</m:r></m:sup><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>D</m:r></m:e><m:sub><m:r>4</m:r></m:sub></m:ssub></m:e></m:spre><m:r>(</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>2</m:r></m:e><m:sup><m:r>n</m:r></m:sup></m:ssup></m:omath> ), where <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>2</m:r></m:e><m:sup><m:r>4</m:r><m:r>n</m:r></m:sup></m:ssup><m:r>-</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>2</m:r></m:e><m:sup><m:r>2</m:r><m:r>n</m:r></m:sup></m:ssup><m:r>+1</m:r></m:omath>  is a prime number by using the order of the group and the largest element order. In order to prove the main theorem, we used Lemmas 4.2, 7.2 of the reference [18]. Results and discussion In this section we prove the main result of this article. For simplicity the Steinberg simple group and prime number are denoted by D and p respectively.  As mentioned in the previous section to prove the main result of this article we use the Lemma 4.2 of [18]. We prove p is an isolated vertex of prime graph. Using Lemma 4.2 we prove that G neither a Frobenius nor 2-Frobenius group. And for the case c this Lemma is satisfied. In other words, G has a normal series such that H and G/K   and K/H are non-abelian simple groups. Moreover, H is a nilpotent group. Every odd components of prime is an odd component of the prime graph.  In the next step, by using Lemma 7.2, since (5, ∣G∣)=1, we consider the groups of this Lemma. We also prove that isomorphism K/H<m:omath><m:r>≇</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>L</m:r></m:e><m:sub><m:r>2</m:r></m:sub></m:ssub></m:omath> (q) , <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>L</m:r></m:e><m:sub><m:r>3</m:r></m:sub></m:ssub></m:omath> (q), <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>U</m:r></m:e><m:sub><m:r>3</m:r></m:sub></m:ssub><m:r>(</m:r><m:r>q</m:r><m:r>)</m:r></m:omath>  , <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>G</m:r></m:e><m:sub><m:r>2</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>q</m:r></m:e></m:d><m:r>, </m:r><m:spre><m:sprepr><m:ctrlpr></m:ctrlpr></m:sprepr><m:sub><m:r> </m:r></m:sub><m:sup><m:r>2</m:r></m:sup><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>G</m:r></m:e><m:sub><m:r>2</m:r></m:sub></m:ssub></m:e></m:spre><m:r>(</m:r><m:r>q</m:r></m:omath> ), are  a contradiction. Finally we have K/H <m:omath><m:r>&cong;</m:r><m:spre><m:sprepr><m:ctrlpr></m:ctrlpr></m:sprepr><m:sub><m:r> </m:r></m:sub><m:sup><m:r>3</m:r></m:sup><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>D</m:r></m:e><m:sub><m:r>4</m:r></m:sub></m:ssub></m:e></m:spre><m:r>(</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>2</m:r></m:e><m:sup><m:r>n</m:r></m:sup></m:ssup></m:omath> ). The proof be completed. Conclusion We conclude that in addition to a previously known criterion(test) for Steinberg groups recognition by their 2-sylow subgroups(<a name="bau1"></a><a href="https://www.sciencedirect.com/science/article/pii/B9780126336504500138#!" style="color:blue; text-decoration:underline">ANTHONY HUGHES</a>, CHARACTERIZATION OF 3D4 (q3), q = 2n  BY ITS SYLOW 2-SUBGROUP, <a href="https://www.sciencedirect.com/science/book/9780126336504" style="color:blue; text-decoration:underline" title="Go to Proceedings of the Conference on Finite Groups on ScienceDirect">Proceedings of the Conference on Finite Groups</a>,1976, Pages 103-105) and also the same order components (Guiyun Chen, Characterization of 3D4(q), <a href="https://link.springer.com/journal/10012" style="color:blue; text-decoration:underline">Southeast Asian Bulletin of Mathematics</a>, 25, pages 389–401,2002). Next, in this paper we can recognize them by the order of the group and the largest element order of the group.   مرتبه ی عناصر, بزرگترین مرتبه ی عناصر, گروه های استینبرگ. element order, the largest elements order, Stienberg group. 108 118 http://mmr.khu.ac.ir/browse.php?a_code=A-10-1421-1&slc_lang=en&sid=1 Ahmad - احمد خاکساری a_khaksari@pnu.ac.ir 10031947532846005993 10031947532846005993 Yes Department of mathematics, Payame noor university, Po.Box:19395-3697 گروه آموزشی ریاضی ، دانشگاه پیام نور، صندوق پستی:19395-3697)