Capability of Nilpotent Products of Cyclic Groups Ii

In Part I it was shown that if G is a p-group of class k, generated by elements of orders 1 < p1 ≤ · · · ≤ pr , then a necessary condition for the capability of G is that r > 1 and αr ≤ αr−1 + ⌊ k−1 p−1 ⌋. It was also shown that when G is the k-nilpotent product of the cyclic groups generated by those elements and k = p = 2 or k < p, then the given conditions are also sufficient. We make a correction related to the small class case, and extend the sufficiency result to k = p for arbitrary prime p. Recall that a group G is said to be capable if and only if there exists a group H such that G ∼= H/Z(H), where Z(H) is the center of H . In [2] we proved that if G is a capable p-group of class k, generated by x1, . . . , xr, with xi of order p αi , 1 ≤ α1 ≤ · · · ≤ αr, then r > 1 and αr ≤ αr−1 + ⌊ k−1 p−1 ⌋. We also proved that if G is the k-nilpotent product of the cyclic p-groups generated by the xi, then the conditions are also sufficient for the cases k < p and k = p = 2. The purpose of this note is twofold: first, we will note an error in a lemma that was used in the proof of the small class case and make the necessary corrections to justify that result. Second, we will extend the result to the case k = p with p an arbitrary prime. Since we follow closely on [2], we refer the reader there for the relevant definitions and conventions. I am extremely grateful to Prof. T. C. Hurley who brought to my attention the results from [1, 7]; these results allowed the correction of the error noted above, as well as simplifying my argument for the k = p case. 1. Shoving commutators In [2], the last clause of Lemma 4.2(ii) is incorrect. Because of this error, the last assertion in Lemma 4.3 is also incorrect; the proof of Theorem 4.4, which describes the center of a k-nilpotent product of cyclic p-groups when k ≤ p, relied on that incorrect assertion and so has a gap. In this section we will provide the necessary correction to justify the conclusion of that theorem. Once it is established, the rest of the proof of the small class case will follow. The error in question is the following: we start with the free group F on x1, . . . , xr, and a basic commutator [u, v] of weight equal to k ≥ 2. Then we considered [u, v, xr]; when v ≤ xr, this is a basic commutator. If v > xr , then we rewrite [u, v, xr] modulo Fk+2 as [u, xr, v][v, xr , u] . The incorrect clause asserted that this expresses [u, v, xr] modulo Fk+2 as a product of basic commutators and their inverses, but this is not necessarily the case; there is no warrant for asserting that [u, xr] or [v, xr ] will necessarily be basic commutators (though they are for small values of k), nor that [v, xr ] > u, another requirement. The main idea is 2000 Mathematics Subject Classification. Primary 20D15, Secondary 20F12.