Frobenius's theorem (group theory)

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

Statement

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

Applications

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

Frobenius's conjecture

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

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

References

• Frobenius, G. (1903), "Über einen Fundamentalsatz der Gruppentheorie", Berl. Ber. (in German): 987–991, doi:10.3931/e-rara-18876, JFM 34.0153.01
• Hall, Marshall (1959), Theory of Groups, Macmillan, LCCN 59005035, MR 0103215
• Iiyori, Nobuo; Yamaki, Hiroyoshi (October 1991), "On a conjecture of Frobenius" (PDF), Bull. Amer. Math. Soc., 25 (2): 413–416, doi:10.1090/S0273-0979-1991-16084-2