On the number of solutions of equation \(x^{p^ k}=1\) in a finite group
Let \(G\) be a finite group, \(N(t, G)\) is the number of solutions of the equation \(x^t=1\) in \(G\). Conditions under which \(N(p^k, G)\) in a finite \(p\)-group is divisible by \(p^{n+k}\) (here \(n\) is a fixed positive integer) are given. It is shown also that congruences for the number of cyclic subgroups of order \(p^k\) which are true for abelian groups hold for more general finite groups (in particular for groups with abelian Sylow \(p\)-subgroups).
groups with Abelian Sylow \(p\)-subgroups
scientific article; zbMATH DE number 817257