Odd Order Group Element is Square/Corollary

Corollary to Odd Order Group Element is Square
Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Then:
 * $\forall x \in G: \exists y \in G: y^2 = x$

iff $\left\vert{G}\right\vert$ is odd.

Proof
Suppose $\left\vert{G}\right\vert$ is odd.

Then from Order of Element Divides Order of Finite Group, all elements of $G$ are of odd order.

Hence:
 * $\forall x \in G: \exists y \in G: y^2 = x$

from Odd Order Group Element is Square.

Now suppose that:
 * $\forall x \in G: \exists y \in G: y^2 = x$

From Odd Order Group Element is Square it follows that all elements of $G$ are of odd order.

Suppose $\left\vert{G}\right\vert$ were even.

Then from Cauchy's Group Theorem it follows that there must exist elements in $G$ of even order.

From that contradiction we conclude that $\left\vert{G}\right\vert$ is odd.