Frobenius's theorem (group theory)

In mathematics, specifically group theory, Frobenius's theorem states that if ${\displaystyle n}$ divides the order of a finite group ${\displaystyle G}$, then the number of solutions of ${\displaystyle x^{n}=1}$ in ${\displaystyle G}$ is a multiple of ${\displaystyle n}$. It was introduced by Frobenius (1903).

A more general version of Frobenius's theorem states that if ${\displaystyle C}$ is a conjugacy class with ${\displaystyle h}$ elements of a finite group ${\displaystyle G}$ with ${\displaystyle g}$ elements and ${\displaystyle n}$ is a positive integer, then the number of elements ${\displaystyle k}$ such that ${\displaystyle k^{n}}$ is in ${\displaystyle C}$ is a multiple of the greatest common divisor ${\displaystyle (hn,g)}$ (Hall 1959, theorem 9.1.1).

One application of Frobenius's theorem is to show that the coefficients of the Artin–Hasse exponential are ${\displaystyle p}$-integral, by interpreting them in terms of the number of elements of order a power of ${\displaystyle p}$ in the symmetric group ${\displaystyle S_{n}}$.

Frobenius conjectured that if, in addition, the number of solutions to ${\displaystyle x^{n}=1}$ is exactly ${\displaystyle n}$, where ${\displaystyle n}$ divides the order of ${\displaystyle G}$, then these solutions form a normal subgroup. This was proved by Iiyori and Yamaki¹ as a consequence of the classification of finite simple groups.

The symmetric group ${\displaystyle S_{3}}$ has exactly ${\displaystyle 4}$ solutions to ${\displaystyle x^{4}=1}$ but these do not form a normal subgroup; this is not a counterexample to the conjecture as ${\displaystyle 4}$ does not divide ${\displaystyle |S_{3}|=6}$.