# Thread: Show G it is a sdubgroup

1. ## Show G it is a sdubgroup

For any x Є G, x-1Hx = {x-1hx: h Є H } is a subgroup of G.

further show that O(H)= O(x-1Hx)

2. ## Re: Show G it is a sdubgroup

Interesting problem. What have you tried?

3. ## Re: Show G it is a sdubgroup

what do you mean by that...???

4. ## Re: Show G it is a sdubgroup

canet anybody solve it....?????

5. ## Re: Show G it is a sdubgroup

Suppose $a,b \in x^{-1}Hx$. Can you show that $ab^{-1} \in x^{-1}Hx$?

(hint: first show that if $b = x^{-1}h'x$ that $b^{-1} = x^{-1}h'^{-1}x$ by multiplying the two together).

Next show that the map:

$h \mapsto x^{-1}hx$ is bijective between $H$ and $x^{-1}Hx$.

(it is fairly obvious it's surjective, so concentrate on proving it's injective).