# Normal Subgroup Conjugate of H by element

1. Feb 9, 2012

### Leb

1. The problem statement, all variables and given/known data
Let H be a subgroup of group G. Then
$H \unlhd G \Leftrightarrow xHx^{-1}=H \forall x\in G$
$\Leftrightarrow xH=Hx \forall x\in G$
$\Leftrightarrow xHx^{-1}=Hxx^{-1} \forall x\in G$
$\Leftrightarrow xHx^{-1}=HxHx^{-1}=H \forall x\in G$
2. Relevant equations
xH={xh:h in H}

3. The attempt at a solution
Reviewed the definitions and properties of cosets/normal subgrps, but could not understand the last step. That is how to get from $Hxx^{-1} to HxHx^{-1}$

EDIT:
Is it simply because $xH=H \Leftrightarrow x \in H$ ?

Last edited: Feb 9, 2012