Let $\displaystyle G$ be a finite group. Prove that if $\displaystyle \mid G \mid$ is odd then every element in $\displaystyle g \in G$ can be expressed as $\displaystyle g=x^2$ for some $\displaystyle x \in G$.
Let $\displaystyle G$ be a finite group. Prove that if $\displaystyle \mid G \mid$ is odd then every element in $\displaystyle g \in G$ can be expressed as $\displaystyle g=x^2$ for some $\displaystyle x \in G$.
Use the pigeon-hole principle. This can be used as $\displaystyle x^2 \neq 1$ for $\displaystyle x \neq 1$, because the order of an element must divide the order of the group, so no element can have order 2.