In mathematical group theory, Frobenius's theorem states that if n divides the order of a finite groupG, then the number of solutions of xn = 1 is a multiple of n. It was introduced by Frobenius (1903).
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, 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 groupSn.
Frobenius conjectureedit
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.
Referencesedit
Frobenius, G. (1903), "Über einen Fundamentalsatz der Gruppentheorie", Berl. Ber.: 987–991, JFM 34.0153.01
Hall, Marshall (1959), Theory of Groups, Macmillan, LCCN 59005035, MR 0103215
November 30, 2023
frobenius, theorem, group, theory, mathematical, group, theory, frobenius, theorem, states, that, divides, order, finite, group, then, number, solutions, multiple, introduced, frobenius, 1903, contents, statement, applications, frobenius, conjecture, reference. 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 Contents 1 Statement 2 Applications 3 Frobenius 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 kn 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 conjecture editFrobenius 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 editFrobenius G 1903 Uber einen Fundamentalsatz der Gruppentheorie Berl Ber 987 991 JFM 34 0153 01 Hall Marshall 1959 Theory of Groups Macmillan LCCN 59005035 MR 0103215 Retrieved from https en wikipedia org w index php title Frobenius 27s theorem group theory amp oldid 1074426259, wikipedia, wiki, book, books, library,
