Odd Order Group Element is Square/Warning

Warning on Odd Order Group Element is Square

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $x \in G$.

Let the order $\order x$ be odd.

Then:

$\exists y \in G: y^2 = x$

False Statement

It is completely false to say:

$\exists y \in G: y^2 = x$

if and only if:

the order $\order x$ is odd

An order $2$ element in $C_4$ refutes the converse.

This mistake can arise by supposing that this:

$\exists y \in G: y^2 = x$

implies:

$\exists n \in \N: \paren {x^n}^2 = x$

The second step can only be used if every $x$ can be expressed in the terms of $y^2$.