Element of a Group

**Markeur** · October 6th 2010, 01:27 AM

Let $G$ be a finite group. Prove that if $\mid G \mid$ is odd then every element in $g \in G$ can be expressed as $g=x^2$ for some $x \in G$ .

**Swlabr** · October 6th 2010, 01:34 AM

Originally Posted by Markeur

Let $G$ be a finite group. Prove that if $\mid G \mid$ is odd then every element in $g \in G$ can be expressed as $g=x^2$ for some $x \in G$ .

Use the pigeon-hole principle. This can be used as $x^2 \neq 1$ for $x \neq 1$ , because the order of an element must divide the order of the group, so no element can have order 2.

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