Question

If every non-identity element of $\mathit{G}$ has order 2, prove that$\mathit{G}$ is abelian.

Short answer

It is proved that $G$ is abelian.

Step-by-step solution

Write the basic properties of G

If $G$ is a group and$a,b\in G$ then,

1. ${\left(ab\right)}^{-1}=b^{-1}{a}^{-1}$;
2. ${\left(a^{-1}\right)}^{-1}=a$

Show that G is an abelian

Assume every no-identity element of $G$has order 2. Then, for$a\ne e$we have $a^2=e$.

Multiply both sides by $a^{-1}$then $a=a^{-1}$.

Assume $a,b\in G$ be two non-identity elements. Then, by assumption${\left(ab\right)}^2=abab=e$.

Multiply by $a$on the left and by $b$on the right. Then we have,

$ba=ab$.

Thus, $G$ is an abelian.

Hence Proved.