The Number of Elements Satisfying $g^5=e$ in a Finite Group is Odd

Group Theory Problems and Solutions in Mathematics

Problem 614

Let $G$ be a finite group. Let $S$ be the set of elements $g$ such that $g^5=e$, where $e$ is the identity element in the group $G$.

Prove that the number of elements in $S$ is odd.

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Proof.

Let $g\neq e$ be an element in the group $G$ such that $g^5=e$.
As $5$ is a prime number, this yields that the order of $g$ is $5$.

Consider the subgroup $\langle g \rangle$ generated by $g$.
As the order of $g$ is $5$, the order of the subgroup $\langle g \rangle$ is $5$.

If $h\neq e$ is another element in $G$ such that $h^5=e$, then we have either $\langle g \rangle=\langle h \rangle$ or $\langle g \rangle \cap \langle h \rangle=\{e\}$ as the intersection of these two subgroups is a subgroup of $\langle g \rangle$.

It follows that $S$ is the union of subgroups of order $5$ that intersect only at the identity element $e$.
Thus the number of elements in $S$ are $4n+1$ for some nonnegative integer $n$.

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