**Let G be a group and a ^{2} = e , for all a **ϵ

**G . Then prove that G is an abelian group.**

**Proof:**

Let a, b ϵG

Then a^{2} = e and b^{2} = e

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

Then (ab)^{2} = e

⇒ (ab)^{2} = a^{2} [Since a^{2} = e]

= (ab)(ab) = a.a

⇒ bab = a [left cancellation law]

Multiply right side by b we get,

bab^{2} = ab

⇒ba.e = ab [Since b^{2} = e]

⇒ ba = ab

Therefore, G is an abelian group.

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