Elements of order 2

Mar 18, 2025 5:52 PM

Suppose $G$ is a finite group of even order.

Prove that an element in $G$ has order dividing 2 if and only if it is its own inverse.
Prove that the number of elements in $G$ of order 2 is odd.
Use (2) to show $G$ must contain a subgroup of order 2.

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L A T E X

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Suppose $G$ is a finite group of even order.
\begin{enumerate}[label=\alph*)]
	\item Prove that an element in $G$ has order dividing 2 if and only if it is its own inverse.
	\item Prove that the number of elements in $G$ of order 2 is odd.
	\item Use (b) to show $G$ must contain a subgroup of order 2.
\end{enumerate}