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 xn = 1 equals n, then the solutions form a normal subgroup.
A more general version of Frobenius's theorem states that if C is a conjugacy class with h elements of a finite groupG 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, theorem 9.1.1).
Applicationsedit
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.
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.
Referencesedit
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
April 13, 2024
frobenius, theorem, group, theory, mathematics, specifically, group, theory, frobenius, theorem, states, that, divides, order, finite, group, then, number, solutions, multiple, introduced, frobenius, 1903, related, frobenius, conjecture, since, proved, frobeni. 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 Contents 1 Statement 2 Applications 3 Frobenius s conjecture 4 ReferencesStatement editA 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 editOne 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 editFrobenius 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 amp 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 editFrobenius G 1903 Uber 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 Retrieved from https en wikipedia org w index php title Frobenius 27s theorem group theory amp oldid 1211479493, wikipedia, wiki, book, books, library,
