Group of even order must contain $a:a=a^{-1}$ $ (a\not = e)$

Question

Let $G$ be a finite group. If the order of $G$ is even, prove that there is at least one element $a$ in $G$ such that $a\not= e$ and $a=a^{-1}$.

Here's my idea:

Suppose $\{x_1,\cdots,x_n\}$ is all of the elements in $G$ [edit: I meant to add "such that $x\not =x^{-1}$"] (but without their inverses or the identity $e$). This set has $n$ elements. Now consider the set of the inverses that correspond to each element: $\{x^{-1}_1,\cdots,x^{-1}_n\}$. Since $G$ is a group, there is a one-to-one correspondence between these two subsets. The total order of these groups is $2n$. Consider the identity $e$, which adds 1 to $\left| G \right|$ to give $G=\{e,x_1,\cdots,x_n,x^{-1}_1,\cdots,x^{-1}_n\}$, which yields: $$|G|=2n+1,$$ which is an odd value. But there may be elements that satisfy $y\in G:y=y^{-1}$. In this case, these elements are redundant and so they each add 1 to the order of $G$. Therefore,

$$|G|=[2n_x+1]+n_y$$

The term in brackets is odd, so $|G|$ is even if and only if the number of elements $y$, or $n_y$, is odd. If $n_y$ is odd, it is clear that $\exists y\in G:y\not = e \wedge y= y^{-1}$. Q.E.D.

My questions are:

1. Is this proof valid?
2. Any recommendations on how I can make this proof shorter and more elegant? (I struggle with wordiness)

Thank you.

Answer 1

Your idea is along the right track, but not worded well.

Note that if any element $g$ in $G$ has even order $2k$, then $g^k=g^{-k}$ is your desired element.

Suppose for sake of contradiction that all non-identity elements of $G$ have odd order. Then as you noted, such an element cannot be its own inverse $g=g^{-1}$, else it would have order $2$. Since inverses are unique, you can group all the non-identity elements of $G$ into pairs. Therefore, $|G|$ is odd, a contradiction.

For a more powerful result, see Cauchy's theorem.

Answer 2

I think you've got the essential idea, but might work a bit on the phrasing. The basic thing to notice (as you have) is that the elements $g$ for which $g\neq g^{-1}$ match up in pairs so, removing those, together with the identiy, from the group leaves an odd number of elements, since the order of the group is even. Hence, the remaining set of elements is, in particular, non-empty! But these are the non-trivial elements $g$ for which $g=g^{-1}$.

Answer 3

Assume $x\neq x^{-1}$ for all $x\neq e$.
Using the equivalence relation $\mathcal R$ in $X=G\backslash\{e\}$ defined by $x\mathcal R y$ iff $xy=e$, each class contain two elements, and this relation give a partition of $X$ to a part (of cardinal two), hence the cardinal of $X$ is even so the order of $G$ is odd (contradiction).