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 is a multiple of n. It was introduced by Frobenius (1903).
Related is Frobenius's conjecture (since proved, but not by Frobenius), which states that if the preceding is true, and the number of solutions of x n = 1 equals n, then the solutions form a normal subgroup.
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 Sn.
This has been proved (Iiyori & Yamaki 1991) as a consequence of the classification of finite simple groups.
The symmetric group S3 has exactly 4 solutions to x4 = 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 S3 which is 6.