TSSFL TECHNOLOGY STACK

Posted: **Tue Jan 05, 2016 1:33 pm**

Problem: Let $G$ be a group. Suppose that $(ab)^2 = a^2b^2 , \ \ \forall \ a, \ b \in G$ , prove that $G$ is abelian.

Proof
$(ab)^{2}(ab)^{-1} = (a^2b^2)(ab)^{-1} \\ \\ (ab)(ab)(ab)^{-1} = (a^2b^2)(ab)^{-1} \\ \\ ab = (a^2b^2)(b^{-1}a^{-1}) = a^2(b^2b^{-1})a^{-1}\\ \\ ab = a^2ba^{-1} \\ \\ aba = a^2b \\ \\ a^{-1}ab = a^{-1}a^2b \\ \\ ba = ab$ .

Hence $G$ is abelian.

TSSFL TECHNOLOGY STACK

Abelian Group - Solved Example

Abelian Group - Solved Example