Question

If $G$ is an abelian group then for all $a,b\in G$ $b^{-1}\ast a^{-1}\ast b\ast a$ is equal to

• A. $a\ast b$
• B. $a^{-1}\ast b^{-1}$
• C. $e$
• D. None of these

Answer 1

The correct option is C $e$

The value of $b^{-1}\ast a^{-1}\ast b\ast a$ is

$=(b^{-1}\ast a^{-1})\ast (b\ast a)$

$={(a\ast b)}^{-1}\ast (b\ast a)$

$={(a\ast b)}^{-1}\ast (a\ast b)=e$