Coset

The powerful notion of Coset was invented by Galois (l'ensemble à gauche ou à droit), but only named as Coset after 150+ yrs later by G.A. Miller in 1910.

Prove:
Coset * Coset = Coset
=> Normal Subgroup

[Hint] Proof technique: use
1) $latex a^{-1}$
2) e

Proof:
1) For any a ∈G, H subgroup of G,

$latex (Ha)(Ha^{-1})= H.(aHa^{-1})$

2) Given $latex H.(aHa^{-1}) $ is right coset,
Choose $latex (aHa^{-1}) = e \in G$
$Latex H.(aHa^{-1})= He = H$
=> $latex aHa^{-1} \subset H$
=> H Normal subgroup