On factorizations of finite Abelian groups which admit replacement of a Z-set by a subgroup (Q1262955)

All groups G considered here are finite and abelian. A subset A of G is a Z-set if for all \(a\in A\) and all \(n\in Z\), na\(\in A\). Let A,B be Z-sets we write \(G=A\oplus B\) if every element \(g\in G\) can be uniquely represented in the form \(g=a+b\), \(a\in A\), \(b\in B\). The paper deals with the problem: Consider the series of G \[ G=G^{(0)}=S\oplus A>G^{(1)}=S^{(1)}\oplus A^{(1)}>...>G^{(n)}=S^{(n)}\oplus A^{(n)}>0 \] where \(S=S^{(0)}\geq S^{(1)}\geq...\geq S^{(n)}\geq (0)\) are subgroups of G and \(A=A^{(0)}\geq A^{(1)}\geq...\geq A^{(n)}\geq \{(0)\}\) are Z-sets, when there exist subgroups \(T^{(i)}\) such that \(G^{(i)}=S^{(i)}\oplus T^{(i)}\) and \(T^{(0)}\geq T^{(1)}\geq...\geq T^{(n)}?\) After reducing the problem to the case when is a G p-group, the author proves: Theorem: Let G be a group of exponent \(p^ k\). Assume \(G=S\oplus A\) and \(G'\) a subgroup of G such that \(G'=S'\oplus A'=S'\oplus T'\) where \(S'\subseteq S\), \(T'\) are subgroups of G and A, \(A'\) Z-sets of G. Then there exists a subgroup T of G verifying \(G=S\oplus T\), \(T\geq T'\) if and only if there exist subgroups \(T_ i\) of G such that \[ T'\geq T_ 1\geq T_ 2\geq...\geq T_ k\geq 0\text{ and }G'\cap p^ iG=(S'\cap p^ iS)\oplus T_ i,\quad 1\geq i\geq k-1. \] This characterization is used to solve the initial problem in some particular cases.