Capability of nilpotent products of cyclic groups II

Abstract

Abstract. In Part I it was shown that if G is a p-group of class k, generated by elements of orders 1 < p α 1 ≤ · · · ≤ p αr , then a necessary condition for the capability of G is that r > 1 and αr ≤ α r−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 x 1 , . . . , x r , with x i of order p αi , 1 ≤ α 1 ≤ · · · ≤ α r , then r > 1 and α r ≤ α r−1 + ⌊ k−1 p−1 ⌋. We also proved that if G is the k-nilpotent product of the cyclic p-groups generated by the x i , 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.