Let G be a group and a^2 = e, for all a ϵG . Then prove that G is an abelian group.

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.

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  1. Pingback: Abelian Group | Breath Math

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