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non-divisibility for abelian groups
Mohammad reza Vedadi 1, Yaser Tolooei2
1- Isfahan University of Technology , mrvedadi@iut.ac.ir
2- Razi University
Abstract:   (1074 Views)
Introduction
  In Throughout all groups are abelian. Suppose that G is a group and n is a positive integer. For a ∈ G, if we consider the solution of the equation nx = a in G, two subsets of G are proposed. One of them is {a ∈ G | ∃x ∈ G,nx = a} and the other is {x ∈ G | nx = a} for given a ∈ 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 ∈< a > then we come to the equation nx = 0 in the group, whose solutions determine a subgroup of   (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 ≥ 1 there exists x ∈ 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. Therefore, G is divisible if and only if nG = G for every n ≥ 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.
Suppose that n ≥ 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 − divisible if there exists a prime number q such that qG ≠ G. Based on the above, we may define the divisibility (non-divisibility) with respect to a number.
Definition 1.1. Let n ≥ 1 a group G is called:
(a)            n-divisible if nG = G.
(b)           Fully non-divisible if pG ≠ G for every prime number p.
(c)            Absolutely non-divisible if pH ≠ H for every prime number p and non-zero subgroup H of G.
Thus, we deal with three class of groups as blow:
{Absolutely non-divisible groups} ⊆ {Fully non-divisible} ∩ {Reduced groups}. Examples are presented to show that these three classes are mutually distinct.
main results
Definition 2.1. For every prime number p, let radp(G) = ∩n≥1pnG and Tp(G), the sum of all p-divisible subgroups of G. Dn be the class of all n-divisible groups and Fp be the class of all groups G with Tp(G) = {0}. Let D={G|G=H⩽GH such that H ∈ ∪n≥1Dn} and Cp be the class of all groups G with radp(G) = {0}.
Theorem 2.2. Let p be a prime number.
    1. For every group homomorphism f: G1 → G2 we have f (radp(G1)) ⊆ radp(G2). Furthermore radp(G) is a fully invariant subgroup of G.
    2. For every H⩽G we have radp(H) ⊆ radp(G). Also if G = H⊕K then radp(H) = H ∩ radp(G).
    3. radp(⊕i∈IGi)=⊕i∈Iradp(Gi).
    4. radp(i∈IGi)=i∈Iradp(Gi).
    5. pG = G if and only if radp(G) = G if and only if  HomZ(G,Zp)={0}.
    6. For every H⩽G we haveradp(G)+HH⊆radp(GH). Also, if H⊆radp(G) thenradp(G)H=radp(GH). FurthermoreGradp(G)={0} .
    7. radp(G)=Rej(G,Cp).
    8. radp(G)=Rej(G,{Zpi}i≥1).
Theorem 2.3. Let p be a prime number.
    1. The class of p-divisible groups is closed under direct sum and homomorphic image.
    2. For every group G, Tp(G) is p-divisible and we have Tp(G) ⊆ radp(G). Furthermore Tp(radp(G)) = radp(Tp(G)) = Tp(G).
    3. If G is a p-torsionfree group, then radp(G) is a p-divisible subgroup and radp(G) = Tp(G).
    4. Let G be a p-torsionfree group and H ≤ G. H ⊆ radp(G) if and only if radp(G) = radp(H). Furthermore radp(radp(G)) = radp(G).
    5. If  Tp(G) = {0}, then p divide the order of every torsion element in G.
    6. Let p and q be two different prime numbers. If Tp(G) = Tq(G) = {0}, then radp(G) = radq(G) = {0}.
    7. Tp(GTp(G))={0}.
Theorem 2.4. For every prime number p, (Dp,Fp) is a torsion theory.
Theorem 2.5. Every absolutely non-divisible group G is torsion free and so G is isomorphic to a subgroup ofQΛ.
Theorem 2.6. The following statements are equivalent for every group G.
    1. G is absolutely non-divisible,
    2. for every prime number p, radq(G) = {0},
    3. HomZ(D,G)={0}.
Theorem 2.7. The class of absolutely non-divisible is closed under direct product and subgroup.
Theorem 2.8. If H and G/H are absolutely non-divisible groups then G is absolutely non-divisible group.
Theorem 2.9. For every group G the following statements hold.
.
(b) G is an absolutely non-divisible group if and only if for every prime number p there exists a natural number n such that  is absolutely non-divisible.
For H⩽Q and prime number p, letBp(H)={t∈N|∃mn∈H,(m,n)=1,pt|n}, and bp(H) = |Bp(H)|.
Theorem 2.10. Let {0} ≠ G ≤ Q G is absolutely non-divisible if and only if for every prime number p, bp(G)<∞.

 
Keywords: Absolutely non-divisible group, fully non-divisible group, p-divisible group, pradical.
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Type of Study: S | Subject: alg
Received: 2020/07/28 | Revised: 2022/11/16 | Accepted: 2021/01/19 | Published: 2022/05/21 | ePublished: 2022/05/21
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