Question

If $(G,\ast )$ is a group such that ${(a\ast b)}^2=(a\ast a)\ast (b\ast b)$ for all $a,b\ast G$, then $G$ is

• A. abelian
• B. finite
• C. infinite
• D. None of these

Answer 1

The correct option is A abelian

${(a\ast b)}^2=(a\ast a)\ast (b\ast b)$ for all $a,b\in G$

$\Rightarrow (a\ast b)\ast (a\ast b)=(a\ast a)\ast (b\ast b)$ for all $a,b\in G$

$a\ast (b\ast a)\ast b=a\ast (a\ast b)\ast b$ for all $a,b\in G$

$\Rightarrow b\ast a=a\ast b$ for all $a,b\in G$ {by cancellation laws}
$\Rightarrow G$ is abelian.