Finite p-Groups With Noninner Automorphisms of Order p

Abstract

Suppose that G is a non-abelian p-group, it was shown that if G is of class 2 then, there exists a noninner automorphism of order p such that C_G(Z(Φ(G))) = Φ(G) [1]. Moreover, if G is of maximal class of order pⁿ, Fouladi S. [13] showed that the order of the group of all automorphisms of G centralizes the Frattini quotient and is not greater than p 2(n−2) if and only if G is metabelian. In this paper, we show that if b(G) = p² and p ≠ 2, then ∩{ker χ | χ(1) = p²} = 1. (Here, b(G) = max(cd(G)) and cd(G)= {χ(1) | χ ∈ Irr(G)}). Suppose further that G is a p-group with Frattini factor group of order ≥ p^2a−1we show that the number of elements of order p in G is congruent to 1 modulo p^a 1 ≤ a ∈ N.