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
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 Sn.
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 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
- 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
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
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 Sn.
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 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
- 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