Let $\displaystyle G$ be a finite group and let $\displaystyle H\neq G$ be a subgroup of $\displaystyle G$. Show that $\displaystyle G\neq \cup_{a\in G} aHa^{-1}$.
Let $\displaystyle G$ be a finite group and let $\displaystyle H\neq G$ be a subgroup of $\displaystyle G$. Show that $\displaystyle G\neq \cup_{a\in G} aHa^{-1}$.
Hint1: Think about orders.
Hint 2: Conjugation of a subgroup preserves order.