On the number of solutions of some Frobenius type equations in finite groups

Let G be a finite group with automorphism group Aut(G) and g is an element of G . For any subgroup H of Aut(G) , define a map xi H:G -> C by xi H(g)=|{(x,alpha)is an element of GxH:[x,alpha]=g}| , where [x,alpha]:=x-1 alpha(x) is called the autocommutator of x and alpha in G. In this paper, we show that xi H is a character of G when H is any subgroup of Aut(G) containing Inn(G) and H=Autcent(G) , where Inn(G) is the inner automorphism group of G and Autcent(G) is the group of all the central automorphisms of G. In these cases expressions for xi H are also obtained in terms of irreducible characters of G. Our results generalize a classical result of Frobenius regarding the number of solutions of the commutator equation in finite groups. As applications of our results, we compute g-autocommuting probabilities of some finite groups and obtained certain bounds for g-autocommuting probability of G in terms of commuting and autocommuting probabilities.