Normal Subgroup Conjugate of H by element

[Leb]

Homework Statement

Let H be a subgroup of group G. Then
[itex]H \unlhd G \Leftrightarrow xHx^{-1}=H \forall x\in G [/itex]
[itex] \Leftrightarrow xH=Hx \forall x\in G [/itex]
[itex] \Leftrightarrow xHx^{-1}=Hxx^{-1} \forall x\in G [/itex]
[itex] \Leftrightarrow xHx^{-1}=HxHx^{-1}=H \forall x\in G [/itex]

Homework Equations

xH={xh:h in H}

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 [itex] Hxx^{-1} to HxHx^{-1}[/itex]

EDIT:
Is it simply because [itex] xH=H \Leftrightarrow x \in H [/itex] ?

 

[mighty2000]

Yes, the last step follows from the fact that if x is an element of H, then xH = H. This can be shown by noting that for any h in H, xh is also in H (since H is a subgroup and x is in H), so xH is a subset of H. Similarly, for any h' in H, x^{-1}h' is also in H, so H is a subset of xH. Therefore, xH = H.