Abelian Group of Order Twice Odd has Exactly One Order 2 Element/Proof 1

Proof
By Abelian Group Factored by Prime, the subgroup $H_2$ defined as:


 * $H_2 := \set {g \in G: g^2 = e}$

has precisely two elements.

One of them has to be $e$, since $e^2 = e$.

The result follows.