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Suppose a [[Finite group|finite group]] $G$ contains a subgroup satisfying specific properties. Using that information, what can be said about the structure of $G$ itself? One way to tackle such a problem is via character theory (cf. also [[Character of a group|Character of a group]]), another is by viewing $G$ as a [[Permutation group|permutation group]]. A classical and beautiful application of character theory is provided in elucidating the structure of Frobenius groups. Namely, let $\{ 1 \} < H < G$. Assume that $H \cap g ^ { - 1 } H g = \{ 1 \}$ whenever $g \in G \backslash H$. Then $H$ is a so-called Frobenius complement in $G$; the group $G$ is then a Frobenius group by definition. It was proved by G. Frobenius in 1901, see [[#References|[a3]]], that the set

\begin{equation*} N = \{ G \backslash ( \bigcup _ { x \in G } x ^ { - 1 } H x ) \} \bigcup \{ 1 \} \end{equation*}

is in fact a [[Normal subgroup|normal subgroup]] of $G$. Almost a century later, Frobenius' proof that $N$ is a subgroup of $G$ is still the only existing proof; it uses character theory! The normal subgroup $N$ is called the Frobenius kernel of $G$.

It can be shown that $G = N H$, that $N \cap H = \{ 1 \}$ and that the orders of $N$ and $H$ are relatively prime. Therefore, by the Schur–Zassenhaus theorem, all Frobenius complements in $G$ are conjugate to each other. Below, let $t$ be an element of a group and let $S$ be a subset of that group; let $C _ { S } ( t )$ denote the set $\{ s \in S : s ^ { - 1 } t s = t \}$.

A finite Frobenius group $G$ with Frobenius complement $H$ and corresponding Frobenius kernel $N$ satisfies:

1) $C _ { G } ( n ) \leq N$ for all $1 \neq n \in N$;

2) $C _ { H } ( n ) = \{ 1 \}$ for all $1 \neq n \in N$;

3) $C _ { G } ( h ) \leq H$ for all $1 \neq h \in H$;

4) every $x \in G \backslash N$ is conjugate to an element of $H$;

5) if $1 \neq h \in H$, then $h$ is conjugate to every element of the coset $Nh$;

6) each non-principal complex irreducible character of $N$ induces irreducibly to $G$.

As a converse, assume that some finite group $G$ contains a normal subgroup $N$ and some subgroup $H$ satisfying $N H = G$ and $N \cap H = \{ 1 \}$. Then the statements 1)–6) are all equivalent to each other, and if one of them is true, then $H$ is a Frobenius complement of $G$, turning $G$ into a Frobenius group with $N$ as corresponding Frobenius kernel. Even more general, if some finite group $G$ with proper normal subgroup $N$ satisfies 1), then, applying one of the [[Sylow theorems|Sylow theorems]], it is not hard to see that all orders of $N$ and $G / N$ are relatively prime. Whence there exists a subgroup $H$ of $G$ satisfying $N H = G$ and $N \cap H = \{ 1 \}$ (by the Schur–Zassenhaus theorem). Thus, again $G$ is a Frobenius group with Frobenius complement $H$ and Frobenius kernel $N$.

Viewed another way, suppose a finite group $G$, containing a non-trivial proper subgroup $H$, acts transitively on a finite set $\Omega$ with $\# \Omega \geq 2$, such that $H = \{ u \in G : \omega ^ { u } = \omega \}$ for some prescribed element $\omega \in \Omega$ and such that only the identity of $G$ leaves invariant more than one element of $\Omega$. Then $G$ is a Frobenius group with Frobenius complement $H$. Any element $n \neq 1$ of the Frobenius kernel $N$ acts fixed-point freely on $\Omega$, i.e. $\omega ^ { n} \neq \omega$ for each $\omega \in \Omega$.

There is a characterization of finite Frobenius groups in terms of group characters only. Namely, let $N$ be a subgroup of a finite group $G$ satisfying $\{ 1 \} < N < G$. Then the following assertions are equivalent:

a) statement 6) above;

b) $N$ is a normal subgroup of $G$ and $G$ is a Frobenius group with Frobenius kernel $N$. The step from b) to a) was known to Frobenius; the converse step with, in addition, $N$ normal in $G$ is surely due to Frobenius; however, the step from a) to b) with $N$ not necessarily normal in $G$ is due to E.B. Kuisch (see [[#References|[a7]]]).

This characterization led Kuisch, and later R.W. van der Waall, to the study of so-called $p$-modular Frobenius groups; see [[#References|[a8]]]. Namely, let $K$ be a [[Field|field]] of positive characteristic $p$. Then $G$ is a $p$-modular Frobenius group if it contains a non-trivial normal subgroup $N$ such that $K$ is a splitting field for the [[Group algebra|group algebra]] $K [ N ]$ and if one of the following (equivalent) statements holds:

A) every non-principal irreducible $K [ N ]$-module $V$ has the property that the induced $K [ G ]$-module $V ^ { G }$ is irreducible;

B) $C _ { G } ( x ) \leq N$ for every $p$-regular non-trivial element $x \in N$. Any $N$ featuring in A)–B) is a $p$-modular Frobenius kernel.

In 1959, J.G. Thompson [[#References|[a9]]] showed that for a  "classical"  Frobenius group $G$, the Frobenius kernel $N$ is nilpotent (cf. also [[Nilpotent group|Nilpotent group]]), thereby solving a long-standing conjecture of W.S. Burnside. It was proved by H. Zassenhaus in 1939, [[#References|[a10]]], that a Sylow $p$-subgroup (cf. also [[Sylow subgroup|Sylow subgroup]]) of a Frobenius complement $H$ of $G$ is cyclic (cf. also [[Cyclic group|Cyclic group]]) when $p$ is odd, and cyclic or generalized quaternion if $p = 2$. He also proved that if $H$ is not solvable (cf. also [[Solvable group|Solvable group]]), then it admits precisely one non-Abelian composition factor, namely the [[Alternating group|alternating group]] on five symbols.

The situation is more involved for $p$-modular Frobenius groups. Namely, a $p$-modular Frobenius kernel $N$ is either solvable (cf. also [[Solvable group|Solvable group]]) or else $p = 2$ and any non–Abelian composition factor of $N$ is isomorphic to $\operatorname { PSL } ( 2,3 ^ { 2^t } )$ for some integer $t \geq 1$.

Furthermore, assume that $N$ is not a [[P-group|$p$-group]]. Then:

any Sylow $q$-subgroup of $G / N$ is cyclic whenever $q$ is relatively prime to $2 p$;

any Sylow $2$-subgroup of $G / N$ is cyclic or generalized quaternion if $p$ is odd. On the other hand, any non-trivial finite $t$-group ($t$ a prime number) is isomorphic to some quotient group $X / Y$, where $X$ is a suitable $t$-modular Frobenius group with $t$-modular Frobenius kernel $Y$. See also [[#References|[a8]]].

Historically, finite Frobenius groups have played a major role in many areas of group theory, notably in the analysis of $2$-transitive groups and finite simple groups (cf. also [[Transitive group|Transitive group]]; [[Simple finite group|Simple finite group]]).

Frobenius groups can be defined for infinite groups as well. Those groups are the non-regular transitive permutation groups in which only the identity has more than one fixed point. Again, let $N$ consist of the identity and those elements of the Frobenius group $G$ not occurring in any point stabilizer (cf. also [[Stabilizer]]). Contrary to the finite case, it is now not always true that $N$ is a subgroup of $G$. See [[#References|[a2]]] for examples.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  Yu.G. Berkovich,   E.M. Zhmud,   "Characters of finite groups" , Amer. Math. Soc.  (1998/9)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J.D. Dixon,   B. Mortimer,   "Permutation groups" , ''GTM'' , '''163''' , Springer  (1996)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  G. Frobenius,   "Ueber auflösbare Gruppen IV"  ''Sitzungsber. Preuss. Akad. Wissenschaft.''  (1901)  pp. 1216–1230</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  B. Huppert,   "Endliche Gruppen" , '''I''' , Springer  (1967)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  B. Huppert,   "Character theory of finite groups" , ''Experim. Math.'' , '''25''' , de Gruyter  (1998)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  I.M. Isaacs,   "Character theory of finite groups" , Acad. Press  (1976)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  E.B. Kuisch,   R.W. van der Waall,   "Homogeneous character induction"  ''J. Algebra'' , '''156'''  (1993)  pp. 395–406</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  E.B. Kuisch,   R.W. van der Waall,   "Modular Frobenius groups"  ''Manuscripta Math.'' , '''90'''  (1996)  pp. 403–427</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  J.G. Thompson,   "Finite groups with fixed point free automorphisms of prime order"  ''Proc. Nat. Acad. Sci. USA'' , '''45'''  (1959)  pp. 578–581</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  H. Zassenhaus,   "Ueber endliche Fastkörper"  ''Abh. Math. Sem. Univ. Hamburg'' , '''11'''  (1936)  pp. 187–220</td></tr></table>