View source for Frobenius conjecture

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In 1903, G. Frobenius published in [[#References|[a1]]] his least known result on finite groups. He proved that if $n$ is a divisor of the order of a [[Finite group|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 [[#References|[a3]]]. In his book [[#References|[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|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|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|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|Sylow theorems]]. In [[#References|[a2]]], M. Hall proved the conjecture for solvable groups (cf. also [[Solvable group|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|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|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 [[#References|[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|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 [[#References|[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 [[#References|[a4]]].

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