# Thread: [SOLVED] Normalizer and Equality of left, right cosets.

1. ## [SOLVED] Normalizer and Equality of left, right cosets.

I think this is a huge assumption to make, but I think this is true. I'm not sure how to bridge the gap.

I'm still kind of confused why for some group H and some $h \in H, aha^{-1} = h' \in H$ with $h' \neq h$ isn't a possibility.

2. What's the huge assumption?

3. Originally Posted by Drexel28
What's the huge assumption?
$aHa^{-1} = H \implies aH = Ha$

Edit: Maybe thats not a huge assumption since

$aHa^{-1} = H \implies aHa^{-1}(a) = H(a) \implies aH = Ha$.

4. Originally Posted by davismj
$aHa^{-1} = H \implies aH = Ha$
$aHa^{-1}=H\implies aH=Ha$?

If $ah\in aH$ then $aha^{-1}\in aHa^{-1}=H$ and thus $aha^{-1}=h'$ for some $h'\in H$ and thus $ah=h'a$ in particular $ah\in Ha$ so that $aH\subseteq Ha$. The reverse inclusion is similar.

5. Originally Posted by Drexel28
$aHa^{-1}=H\implies aH=Ha$?

If $ah\in aH$ then $aha^{-1}\in aHa^{-1}=H$ and thus $aha^{-1}=h'$ for some $h'\in H$ and thus $ah=h'a$ in particular $ah\in Ha$ so that $aH\subseteq Ha$. The reverse inclusion is similar.
Thanks bud.