نابخش‌پذیری گروه‌های آبلی

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 rad_p(G) = ∩_n≥1pⁿG and T_p(G), the sum of all p-divisible subgroups of G. D_n be the class of all n-divisible groups and F_p be the class of all groups G with T_p(G) = {0}. Let D={G|G=H⩽GH such that H ∈ ∪_n≥1D_n} and C_p be the class of all groups G with rad_p(G) = {0}.

Theorem 2.2. Let p be a prime number.



For every group homomorphism f: G₁ → G₂ we have f (rad_p(G₁)) ⊆ rad_p(G₂). Furthermore rad_p(G) is a fully invariant subgroup of G.
For every H⩽G we have rad_p(H) ⊆ rad_p(G). Also if G = H⊕K then rad_p(H) = H ∩ rad_p(G).
radp(⊕i∈IGi)=⊕i∈Iradp(Gi).
radp(i∈IGi)=i∈Iradp(Gi).
pG = G if and only if rad_p(G) = G if and only if HomZ(G,Zp)={0}.
For every H⩽G we haveradp(G)+HH⊆radp(GH). Also, if H⊆radp(G) thenradp(G)H=radp(GH). FurthermoreGradp(G)={0} .
radp(G)=Rej(G,Cp).
radp(G)=Rej(G,{Zpi}i≥1).



Theorem 2.3. Let p be a prime number.




The class of p-divisible groups is closed under direct sum and homomorphic image.
For every group G, T_p(G) is p-divisible and we have T_p(G) ⊆ rad_p(G). Furthermore T_p(rad_p(G)) = rad_p(T_p(G)) = T_p(G).
If G is a p-torsionfree group, then rad_p(G) is a p-divisible subgroup and rad_p(G) = T_p(G).
Let G be a p-torsionfree group and H ≤ G. H ⊆ rad_p(G) if and only if rad_p(G) = rad_p(H). Furthermore rad_p(rad_p(G)) = rad_p(G).
If T_p(G) = {0}, then p divide the order of every torsion element in G.
Let p and q be two different prime numbers. If T_p(G) = T_q(G) = {0}, then rad_p(G) = rad_q(G) = {0}.
Tp(GTp(G))={0}.



Theorem 2.4. For every prime number p, (D_p,F_p) 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.




G is absolutely non-divisible,
for every prime number p, rad_q(G) = {0},
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 b_p(H) = |B_p(H)|.

Theorem 2.10. Let {0} ≠ G ≤ Q G is absolutely non-divisible if and only if for every prime number p, b_p(G)<∞.