Breath Math

Let G be a group and a² = e , for all a ϵG . Then prove that G is an abelian group.

Proof:

Let a, b ϵG

Then a² = e and b² = e

Since G is a group, a , b ϵ G [by associative law]

Then (ab)² = e

⇒ (ab)² = a² [Since a² = e]

= (ab)(ab) = a.a

⇒ bab = a [left cancellation law]

Multiply right side by b we get,

bab² = ab

⇒ba.e = ab [Since b² = e]

⇒ ba = ab

Therefore, G is an abelian group.