Central decompositions of finite \(p\)-groups with Abelian second center and with center of order \(p\) (Q1078659)

Let \(G\) be a finite group and \(U_ 1,...,U_ r\) subgroups of \(G\) such that (i) \(G\) is generated by \(U_ 1,...,U_ r\) (ii) \(U_ i,U_ j\) (\(i\neq j\)) commute elementwise and (iii) \(Z(G)\subseteq U_ i\), \(i=1,2,...,r\). Then the factorization \(G=U_ 1U_ 2...U_ r\) is called a central decomposition of \(G\). The decomposition is nondecomposable if none of the subgroups \(U_ i\) possesses a non-trivial central decomposition. Two nondecomposable central decompositions (NCDs) are said to be isomorphic if the corresponding direct sum decompositions of \(G/Z(G)\) are isomorphic (in the sense of Krull-Remak-Schmidt). If \(G\) is a \(p\)-group and \(G=U_ 1\cdots U_ r\) is an NCD as above and the second centre \(Z_ 2(U_ i)\) has order \(p^ 2\) for \(i=1,2,...,r\), the author shows that \(G\) has precisely \(p^{r(r-1)/2}\) NCDs and that if \(p>2\), they are all isomorphic. Examples are given to show the result can fail for \(p=2\) and a modified result is proved.