Frobenius conjecture on nth roots

Statement

Suppose $G$ is a finite group and $n$ is a natural number dividing the order of $G$. Suppose the number of $n^{th}$ roots in $G$, i.e., the number of elements $g\in G$ such that $g^{n}=e$, is exactly $n$.

The Frobenius conjecture on nth roots states that in that case, that set of $n^{th}$ roots must be a subgroup of $G$.

Related facts

• Number of nth roots is a multiple of n
• Number of nth roots of any conjugacy class is a multiple of n
• At most n elements of order dividing n implies every finite subgroup is cyclic