Hi there vincent!

Swlabr : consider $\displaystyle S_3$. Each of $\displaystyle (12), (13), (23)$ has order $\displaystyle 2$; the other three elements $\displaystyle 1, (123), (132)$ form a subgroup isomorphic to the cyclic group of three elements. So that's another example.

It is not true that the product of two elements of even order has even order. For example $\displaystyle (12)(23)=(123)$ has order $\displaystyle 3$.

Also you say but there could be an element of order not $\displaystyle 2$ which is not of odd order!

Vincent : are you

*given* that the elements of order other than $\displaystyle 2$ form a subgroup? Because in general that's not true. For example in $\displaystyle Z_4=<x>$, $\displaystyle x$ has order $\displaystyle 4$ (which is not $\displaystyle 2$) but $\displaystyle x^2$ has order 2.