Problem

Source: Romania National Olympiad 2014, Grade XII, Problem 4

Tags: number theory, prime numbers, group theory, abstract algebra



Let be a finite group $ G $ that has an element $ a\neq 1 $ for which exists a prime number $ p $ such that $ x^{1+p}=a^{-1}xa, $ for all $ x\in G. $ a) Prove that the order of $ G $ is a power of $ p. $ b) Show that $ H:=\{x\in G|\text{ord} (x)=p\}\le G $ and $ \text{ord}^2(H)>\text{ord}(G). $