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.

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.

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