View source for Frobenius's theorem (group theory)

In mathematical group theory, '''Frobenius's theorem''' states that if ''n'' divides the order of a [[Finite group|finite group]] ''G'', then the number of solutions of ''x''<sup>''n''</sup> = 1 is a multiple of ''n''. It was introduced by {{harvs|txt|last=Frobenius|year=1903|}}.
==Statement==

A more general version of Frobenius's theorem states that if ''C'' is a [[Conjugacy class|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''<sup>''n''</sup> is in ''C'' is a multiple of the greatest common divisor (''hn'',''g'') {{harv|Hall|1959|loc=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|symmetric group]] ''S''<sub>''n''</sub>.

==Frobenius conjecture==

Frobenius conjectured that if in addition the number of solutions to ''x''<sup>''n''</sup>=1 is exactly ''n'' where ''n'' divides the order of ''G'' then these solutions form a [[Normal subgroup|normal subgroup]]. This has been proved as a consequence of the [[Classification of finite simple groups|classification of finite simple groups]]. The symmetric group ''S''<sub>3</sub> has exactly 4 solutions to ''x''<sup>4</sup>=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 ''S''<sub>3</sub>.

==References==

*{{citation|last=Frobenius|first= G.
|title=Über einen Fundamentalsatz der Gruppentheorie| JFM =34.0153.01
|journal=Berl. Ber.|year= 1903|pages= 987–991}}
* {{citation | first1=Marshall | last1=Hall | title=Theory of Groups | publisher=Macmillan | year=1959 |mr=103215 | lccn=59005035 }}
[[Category:Theorems in group theory]]

{{Sourceattribution|Frobenius's theorem (group theory)}}