Groups of prime power order with derived subgroup of prime order (Q1305446)

A parametrization of groups of order \(p^n\) (\(p\) prime, \(n\geq 3\)) with commutator subgroup of order \(p\) is given. The set of parameters is independent of \(p\), although the proof for \(p=2\) is different from the odd case. Along the way a parametrization of the orbits of the general symplectic group on the set of flags of the underlying vector space is also obtained. Slightly reformulated, the main result establishes a bijection between the isomorphism classes of these groups and the following systems of nonnegative integers \((c,m_i,e,\alpha_{ij})\) such that \(c<n\), \(n-c\) is even, \(\sum im_i=c\), \(m_e>0\), \(\sum_{1\leq i\leq j\leq c+2}\alpha_{ij}=(n-c)/2\), and for all \(k\), \(2\leq k\leq c+2\) the sum \(\sum_{1\leq i\leq k}\alpha_{ik}+\sum_{k\leq j\leq c+2}\alpha_{kj}\) is at most \(m_{k-1}\), if \(k\leq e\); \(1\), if \(k=e+1\); \(m_e-1\), if \(k=e+2\); \(m_{k-2}\), if \(k>e+2\). The defining relations of the group corresponding to this system of parameters is too lengthy to be reproduced here (see Proposition 9 in the paper). We just note here that \(|Z(G)|=p^c\), \(m_i\) is the number of direct factors of order \(p^i\) in the canonical decomposition of \(Z(G)\), \(e\) is the largest integer such that \(G'\subseteq Z(G)^{p^{e-1}}\), and the \(\alpha_{ij}\) give the parametrization of a flag of \(G/Z(G)\) considered as a symplectic space.