On the number of solutions of \(x^{p^ k}=a\) in a p-group (Q1095252)

This paper studies the problem of determining the p-divisibility of the number of solutions of the equation \(x^{p^ k}=a\) in a finite group. The following result is obtained. Theorem. Let G be a finite group, and \(H\subseteq G\) be a p-elementary abelian subgroup of order \(p^ r\). Then, for any central element \(a\in Z(G)\) and any integer \(k\geq 1\), the number N of solutions of the equation \(x^{p^ k}=a\) in G is divisible by \(p^{r-[r/p^ k]}\). ([d] denotes the integral part of a positive number d.) In particular, N is divisible by \(p^{\min (r,p^ k-1)}\). This theorem is, in general, not true is a is not assumed central. However, it has the following variant if a is assumed to be a p-element instead. Theorem. Let G, H and r be as above. Then, for any p-element \(a\in G\), the number of solutions of the equation \(x^{p^ k}=a\) in G is divisible by \(p^{r-1}\) when \(p^ k=r\), and divisible by \(p^ r\) when \(p^ k>r.\) Specializing this to p-groups, one obtains the following two results which generalize earlier theorems of Kulakoff, Huppert and Berkovich. Theorem. Let G be a finite non-cyclic p-group, where \(p>2\). Let \(a\in G\) and k be any positive integer. Then the number of solutions of the equation \(x^{p^ k}=a\) in G is divisible by \(p^ 2\). Theorem. Let G be a finite p-group which is not metacyclic, \(a\in G\), and k be a positive integer. Assume that (i) \(p>3\), or (ii) \(p=3\), \(k\geq 2\), and G is not one of the 3-groups determined by Blackburn which do not have a normal subgroup \(\cong {\mathbb{Z}}_ 3\oplus {\mathbb{Z}}_ 3\oplus {\mathbb{Z}}_ 3\). Then the number of solutions of the equation \(x^{p^ k}=a\) in G is divisible by \(p^ 3\). The proofs of the above results made use of the modular representation theory of a cyclic p-group.