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) a^{-1}
2) e

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

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

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

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