Frobenius's theorem (group theory)
# Frobenius's theorem (group theory)

In mathematical 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 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 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 has been proved 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 $S_3.$

