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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