Odd Order Group Element is Square/Warning

False Statement
It is completely false to say:


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


 * the order $\order x$ is odd
 * 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 $y^2$.