A Study of the Number of Roots of xk = g in a Finite Group via Its Frobenius-Schur Indicators

Let G be a finite group. For k epsilon N, x epsilon Irr(G), define e(x)((k))) := 1/broken vertical bar C vertical bar Sigma(g epsilon G)x(g(k)). This is called the k-th Frobenius-Schur indicator of x. In this article we study the Frobenius-Schur indicators for Frobenius groups, p-groups, semidihedral groups and modular p-groups. Further, we use this to study the function (sigma(k)(G)(g) which counts the number of roots of x(k) = g in a finite group G.