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.