Smooth \(p\)-groups (Q5917345)

A chain of subgroups \(1=G_0<G_1<\cdots<G_n=G\) in a group \(G\) is said to be smooth if each \(G_i\) is maximal in \(G_{i+1}\), and all intervals \([G_{i+k}/G_i]=\{H\leq G:G_i\leq H\leq G_{i+k}\}\) are isomorphic as partially ordered sets, for \(i+k\leq n\). If \(G\) is a finite \(p\)-group, the author has shown [Geom. Dedicata 84, No. 1-3, 183-206 (2001; Zbl 0985.20012)] that all \(G_i\) are normal in \(G\); the chain is said to be strongly smooth if all factor groups \(G_{i+k}/G_i\) are isomorphic, for \(i+k\leq n\). A group is said to be (strongly) smooth if it has a (strongly) smooth chain. \textit{H.~Schnabel} [Arch. Math. (to appear)] has proved that if \(|\Omega_1(G_m)|=p^r\) in the strongly smooth finite \(p\)-group \(G\), then \(G\) has order at most \(p^{r-1}|G_m|\), and nilpotency class at most two, provided \(p\) is odd if \(m<2r\), and \(G_{m+1}\) has exponent \(p^2\) if \(G_m\) is elementary Abelian. In the paper under review bounds on the order of \(G\) are proved in the cases not covered by Schnabel. For instance, if \(G_m\) is elementary Abelian, then \(G\) has order at most \(p^{(s+1)m}\), where \(s=\lceil\log_2(m)\rceil\). For each prime \(p\) and infinitely many \(m\), smooth groups \(G\) of order \(p^{2m}\) are constructed in which \(G_m\) is elementary Abelian; thus Schnabel's bound does not carry over here.