Frobenius conjecture

In 1903, G. Frobenius published in [a1] his least known result on finite groups. He proved that if $n$ is a divisor of the order of a finite group $G$, then the number of solutions of $x^n=1$ in $G$ is a multiple of $n$. This result was greatly generalized by Ph. Hall in [a3]. In his book [a2], M. Hall proved the following generalization of Frobenius' theorem: If $G$ is a finite group of order $g$ and $C$ is a conjugacy class of $G$ of cardinality $h$, then the number of solutions of $x^n=c$ in $G$, when $c$ ranges over $C$, is a multiple of the greatest common divisor $(hn,g)$.

The Frobenius conjecture deals with a special case of the result proved by Frobenius. It claims that if $n$ is a divisor of the order of the finite group $G$ and if the number of solutions of $x^n=1$ in $G$ is exactly $n$, then these solutions form a normal subgroup of $G$. It is clear that one needs only to prove the closure of the set of solutions. Thus, the conjecture holds in Abelian groups (cf. also Abelian group). It is also easy to see that it suffices to show that $G$ contains a subgroup of order $n$. Hence, the conjecture certainly holds whenever $n=p^r$, a power of a prime number, since $G$ contains a subgroup of order $n$ by one of the Sylow theorems. In [a2], M. Hall proved the conjecture for solvable groups (cf. also Solvable group). Still, the general problem remained open for a long period and it was solved only recently (1998), using the classification of the finite simple groups (cf. also Simple finite group). It is worthwhile to mention that the assumption that $n$ is a divisor of the order of $G$ is essential. Thus, for example, $x^4=1$ has exactly $4$ solutions in the symmetric group on three letters, but obviously the solutions do not form a subgroup of $G$.

Connection with the classification problem.

It was shown in 1954 by R.A. Zemlin in his PhD thesis [a6] that it suffices to prove the conjecture for non-Abelian simple groups. In other words, one needs to prove that if $G$ is a simple group and $n$ is a divisor of $|G|$, then the number of solutions of $x^n=1$ equals $n$ only for the trivial values of $n$: $n=1$ or $n=|G|$. In [a5] M. Murai proved the same result and showed, in addition, that it suffices to consider those divisors $n$ of $|G|$ which satisfy $(n,|G|/n)=1$.

The conjecture has been verified for the alternating groups, the sporadic groups and the finite simple groups of Lie type by M.J. Rust, H. Yamaki and N. Iiyori in a long series of papers, the last and concluding one being [a4].

References

[a1]	G. Frobenius, "Über einen Fundamentalsatz der Gruppentheorie" Berl. Sitz. (1903) pp. 987–991
[a2]	M. Hall, "The theory of groups" , Macmillan (1959)
[a3]	P. Hall, "On a theorem of Frobenius" Proc. London Math. Soc. , 7 : 3 (1956) pp. 1–42
[a4]	N. Iiyori, "A conjecture of Frobenius and the simple groups of Lie type, IV" J. Algebra , 154 (1993) pp. 188–214
[a5]	M. Murai, "On the Frobenius conjecture" SÛgaku , 35 (1983) pp. 82–84 (In Japanese)
[a6]	R.A. Zemlin, "On a conjecture arising from a theorem of Frobenius" PhD Thesis Ohio State Univ. (1954)