# Math Help - Help!

1. ## Help!

(Let G be a group and let H ≤ G . If for all x ∈ G, (x^2)ϵH, then show that H is a normal subgroup of G,
G/H is commutative quotient group.
$\forall x \in G, \ \forall h \in H: \ xhx^{-1}=h^{-1}(hx)^2(x^{-1})^2 \in H. \ \ \ \square$
since $Hg=Hg^{-1}, \ \forall g \in G,$ for any $x,y \in G$ we'll have: $HxHy=Hxy=H(xy)^{-1}=Hy^{-1}x^{-1}=Hy^{-1}Hx^{-1}=HyHx. \ \ \square$