Frobenius's theorem (group theory)

In mathematics, specifically group theory, Frobenius's theorem states that if ${\displaystyle n}$ divides the order of a finite group ${\displaystyle G}$, then the number of solutions of ${\displaystyle x^{n}=1}$ in ${\displaystyle G}$ is a multiple of ${\displaystyle n}$. It was introduced by Frobenius  ( 1903 ).

Statement

A more general version of Frobenius's theorem states that if ${\displaystyle C}$ is a conjugacy class with ${\displaystyle h}$ elements of a finite group ${\displaystyle G}$ with ${\displaystyle g}$ elements and ${\displaystyle n}$ is a positive integer, then the number of elements ${\displaystyle k}$ such that ${\displaystyle k^{n}}$ is in ${\displaystyle C}$ is a multiple of the greatest common divisor ${\displaystyle (hn,g)}$( 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 ${\displaystyle p}$-integral, by interpreting them in terms of the number of elements of order a power of ${\displaystyle p}$ in the symmetric group ${\displaystyle S_{n}}$.

Frobenius's conjecture

Frobenius conjectured that if, in addition, the number of solutions to ${\displaystyle x^{n}=1}$ is exactly ${\displaystyle n}$, where ${\displaystyle n}$ divides the order of ${\displaystyle G}$, then these solutions form a normal subgroup. This was proved by Iiyori and Yamaki ^[1] as a consequence of the classification of finite simple groups.

The symmetric group ${\displaystyle S_{3}}$ has exactly ${\displaystyle 4}$ solutions to ${\displaystyle x^{4}=1}$ but these do not form a normal subgroup; this is not a counterexample to the conjecture as ${\displaystyle 4}$ does not divide the order of ${\displaystyle S_{3}}$, which is ${\displaystyle 6}$.

Notes

1. 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.

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