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 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.
Statement
[edit]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
[edit]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.
Frobenius's conjecture
[edit]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 (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.
References
[edit]- 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