On the wreath product of Z-groups (Q2639176)

A Z-group is a group having a central series (central system), of arbitrary order type. This paper gives some sufficient conditions for the wreath product of two Z-groups to be a Z-group. Some previous work on this problem is cited in the paper; note also that \textit{A. Lichtman} [Isr. J. Math. 26, 276-293 (1977; Zbl 0365.20009)] gave necessary and sufficient conditions for the residual nilpotence of the augmentation ideal. The two main results (Theorems 2 and 3) are actually false as stated. Theorem 3 states that the free product of any set of Z-groups is a Z- group. However if p and q are distinct primes, then the free product of a cyclic group of order p and one of order q is not a Z-group. If the two cyclic groups are generated by x and y respectively, and G is their free product, there can be no central factor U/V of G such that [x,y]\(\in U\setminus V.\) The statement of Theorem 2 requires the notion of the central spectrum \(\sigma\) (G) of a Z-group G. If \(1\neq x\in G\) and X is the normal closure of x in G, then X/[X,G] is a nontrivial cyclic group. The set \(\sigma\) (G) consists of all prime divisors of the orders of all such groups that are finite, together with 0 if X/[X,G] is finite cyclic for some \(x\in G\). Theorem 2 states the following. Let A and G be Z-groups and suppose one of the following holds: 1) \(\sigma (A)=\{0\}\); 2) \(\sigma (G)=\{0\}\); 3) \(\sigma (A)=\{0,p\}\) and \(\sigma (G)=\{0,p\}\) where p is a prime. Then the wreath product of \(A\wr G\) is a Z-group and \(\sigma\) (A\(\wr G)=\sigma (A)\cup \sigma (G)\). This is certainly not always true under hypothesis 1); for example it is false if A is infinite cyclic and G is a cyclic group of order pq for distinct primes p and q. It is claimed in case 3) that both G and A have central series with factors of order p, and a similar assertion plays a role in case 2). The reviewer has not been able to convince himself of this, but if one makes this as a hypothesis in case 3), and replaces case 2) by one in which G has a central series with torsion free factors, then the proofs appear to go through. The proofs then involve a reasonably straightforward reduction to the case when A is cyclic of order p and G has a central series with factors of order p, when with some work an explicit central series can be constructed in the wreath product from one in G. (Theorem 1). A somewhat easier but essentially equivalent proof can also be given using a criterion due to \textit{K. Hickin} and \textit{R. Phillips} [Arch. Math. 24, 346-350 (1973; Zbl 0275.20070), Corollary to Theorem 1].