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 xn = 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 kn is in C is a multiple of the greatest common divisor (hn,g) (Hall 1959).
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 Sn.
Frobenius conjecture
Frobenius conjectured that if in addition the number of solutions to xn=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 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.
References
- Frobenius, G. (1903), "Über einen Fundamentalsatz der Gruppentheorie", Berl. Ber.: 987–991
- Hall, Marshall (1959), Theory of Groups, Macmillan
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