On the number of \(p\)-subgroups of a finite group
Let \(G\) be a finite group. For each positive integer \(n\), put \(G(n)=\{x\in G\mid x^n=1\}\) and \(m(G,n)=|G(n)|/(n,|G|)\). Let \(p\) be a prime. For each integer \(e\), let \(S_e(G)\) be the set of subgroups of \(G\) of order \(p^e\) and put \(s_e(G)=|S_e(G)|\). Let \(P\) be a Sylow \(p\)-subgroup of \(G\) of order \(p^n\). Frobenius has proved that \(m(G,n)\) is always an integer and \(s_e(G)\equiv 1\bmod p\) for \(0\leq e\leq n\). Kulakov has proved that if \(p\) is odd and \(P\) is not cyclic then \(s_e(G)\equiv 1+p\bmod p^2\) for \(1\leq e\leq n-1\) and \(m(G,p^e)\) is a multiple of \(p\) for \(1\leq e\leq n-1\). A generalization of Kulakov's result is given here. The author calls a \(p\)-group \(P\) exceptional if \(P\) is cyclic (\(p\neq 2\)) and if \(P\) is cyclic, generalized quaternion of order \(\geq 8\), dihedral or semi-dihedral (\(p=2\)). For a family \(X\) of \(p\)-groups, a group \(G\) and an integer \(e\), let \(S_e(G,X)=\{H\mid H\leq G,\;|H|=p^e,\;H\in X\}\) and put \(s_e(G,X)=|S_e(G,X)|\). It is proved that if \(G\) is a group with a Sylow \(p\)-subgroup \(P\) of order \(p^n\) (\(n\geq 2\)) and if \(P\) is non-exceptional then \(s_e(G)\equiv 1+p\bmod p^2\) for any \(e\) with \(1\leq e\leq n-1\); if \(P\) is exceptional \(s_e(G)\equiv 1\bmod p^2\) for any \(e\) with \(1\leq e\leq n-2\) and \(s_{n-1}(G)\equiv 1\) or \(1+p\bmod p^2\) according as \(P\) is cyclic or not.
scientific article; zbMATH DE number 1910468