Let G be a group and a, b ϵ G with O(a) = 5, a3b = ba3 .Prove that G is an abelian group.
Proof:
Given a3b = ba3
Multiply left side by a2,
a2.a3.b = a2.b.a3
a5.b = a2.b.a3
1.b = a2.b.a3 [since O(a) = 5 i.e., a5 = 1]
Multiply right side by a2,
ba2 = a2ba3a2
ba2 = a2ba5 [since O(a) = 5 i.e., a5 = 1]
ba2 = a2b
Multiply right side by a
ba3 = a2ba
a3b = a2ba [since a3b = ba3]
a2.a.b = a2b.a
ab = ba [by left cancellation law]
Therefore, G is abelian group.
Breath Math 
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[…] Let G be a group and a, b ϵ G with O(a) = 5, a^3b = ba^3 .Prove that G is an abelian group. […]
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