1. ## Finite Group Question

Let $\displaystyle G$ be a finite group and let $\displaystyle H\neq G$ be a subgroup of $\displaystyle G$. Show that $\displaystyle G\neq \cup_{a\in G} aHa^{-1}$.

2. Originally Posted by paulk
Let $\displaystyle G$ be a finite group and let $\displaystyle H\neq G$ be a subgroup of $\displaystyle G$. Show that $\displaystyle G\neq \cup_{a\in G} aHa^{-1}$.

Hint 2: Conjugation of a subgroup preserves order.

3. Let $\displaystyle [G:H]=n\geq 2$. Then we have $\displaystyle |\cup_{a\in G} aHa^{-1}|\leq n(|H|-1)+1<n|H|=|G|$.