# Math Help - G cannot be the union of conjugates

1. ## G cannot be the union of conjugates

If $G$ is a finite group and $H$ is a subgroup of $G$ then prove that $\displaystyle G \neq \bigcup\limits_{a \in G} aHa^{-1}$

2. Originally Posted by Chandru1
If $G$ is a finite group and $H$ is a subgroup of $G$ then prove that $\displaystyle G \neq \bigcup\limits_{a \in G} aHa^{-1}$
Of course, we assume H is a PROPER subgroup.

Let G act by conjugation on the set X of all its proper subgroups; then we get that

$s:=|Orb(H)|=[G:N_G(H)]\leq [G:H]=r$ , say, and since $|aHa^{-1}|=|H|\,\,\,\forall\,a\in G$ ,we get that:

$|\bigcup aHa^{-1}|\leq 1+s(|H|-1)\leq 1+r(|H|-1)=1+r|H|-r=|G|-(r-1)<|G|$

(Question: why $|\bigcup aHa^{-1}|\leq 1+s(|H|-1)$ ??)

Tonio