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Suppose a [[Finite group|finite group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f1201901.png" /> contains a subgroup satisfying specific properties. Using that information, what can be said about the structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f1201902.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f1201903.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f1201904.png" />. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f1201905.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f1201906.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f1201907.png" /> is a so-called Frobenius complement in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f1201908.png" />; the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f1201909.png" /> is then a Frobenius group by definition. It was proved by G. Frobenius in 1901, see [[#References|[a3]]], that the set
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019010.png" /></td> </tr></table>
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is in fact a [[Normal subgroup|normal subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019011.png" />. Almost a century later, Frobenius' proof that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019012.png" /> is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019013.png" /> is still the only existing proof; it uses character theory! The normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019014.png" /> is called the Frobenius kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019015.png" />.
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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 $\{ 1 \} &lt; H &lt; G$. 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
  
It can be shown that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019016.png" />, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019017.png" /> and that the orders of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019019.png" /> are relatively prime. Therefore, by the Schur–Zassenhaus theorem, all Frobenius complements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019020.png" /> are conjugate to each other. Below, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019021.png" /> be an element of a group and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019022.png" /> be a subset of that group; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019023.png" /> denote the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019024.png" />.
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\begin{equation*} N = \{ G \backslash ( \bigcup _ { x \in G } x ^ { - 1 } H x ) \} \bigcup \{ 1 \} \end{equation*}
  
A finite Frobenius group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019025.png" /> with Frobenius complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019026.png" /> and corresponding Frobenius kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019027.png" /> satisfies:
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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 $G$.
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019028.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019029.png" />;
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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 $\{ s \in S : s ^ { - 1 } t s = t \}$.
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019030.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019031.png" />;
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A finite Frobenius group $G$ with Frobenius complement $H$ and corresponding Frobenius kernel $N$ satisfies:
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019032.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019033.png" />;
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1) $C _ { G } ( n ) \leq N$ for all $1 \neq n \in N$;
  
4) every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019034.png" /> is conjugate to an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019035.png" />;
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2) $C _ { H } ( n ) = \{ 1 \}$ for all $1 \neq n \in N$;
  
5) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019036.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019037.png" /> is conjugate to every element of the coset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019038.png" />;
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3) $C _ { G } ( h ) \leq H$ for all $1 \neq h \in H$;
  
6) each non-principal complex irreducible character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019039.png" /> induces irreducibly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019040.png" />.
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4) every $x \in G \backslash N$ is conjugate to an element of $H$;
  
As a converse, assume that some finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019041.png" /> contains a normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019042.png" /> and some subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019043.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019045.png" />. Then the statements 1)–6) are all equivalent to each other, and if one of them is true, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019046.png" /> is a Frobenius complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019047.png" />, turning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019048.png" /> into a Frobenius group with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019049.png" /> as corresponding Frobenius kernel. Even more general, if some finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019050.png" /> with proper normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019051.png" /> satisfies 1), then, applying one of the [[Sylow theorems|Sylow theorems]], it is not hard to see that all orders of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019053.png" /> are relatively prime. Whence there exists a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019054.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019055.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019057.png" /> (by the Schur–Zassenhaus theorem). Thus, again <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019058.png" /> is a Frobenius group with Frobenius complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019059.png" /> and Frobenius kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019060.png" />.
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5) if $1 \neq h \in H$, then $h$ is conjugate to every element of the coset $Nh$;
  
Viewed another way, suppose a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019061.png" />, containing a non-trivial proper subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019062.png" />, acts transitively on a finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019063.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019064.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019065.png" /> for some prescribed element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019066.png" /> and such that only the identity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019067.png" /> leaves invariant more than one element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019068.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019069.png" /> is a Frobenius group with Frobenius complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019070.png" />. Any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019071.png" /> of the Frobenius kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019072.png" /> acts fixed-point freely on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019073.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019074.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019075.png" />.
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6) each non-principal complex irreducible character of $N$ induces irreducibly to $G$.
  
There is a characterization of finite Frobenius groups in terms of group characters only. Namely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019076.png" /> be a subgroup of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019077.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019078.png" />. Then the following assertions are equivalent:
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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 $N$.
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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 $\omega \in \Omega$.
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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 $\{ 1 \} &lt; N &lt; G$. Then the following assertions are equivalent:
  
 
a) statement 6) above;
 
a) statement 6) above;
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019079.png" /> is a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019081.png" /> is a Frobenius group with Frobenius kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019082.png" />. The step from b) to a) was known to Frobenius; the converse step with, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019083.png" /> normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019084.png" /> is surely due to Frobenius; however, the step from a) to b) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019085.png" /> not necessarily normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019086.png" /> is due to E.B. Kuisch (see [[#References|[a7]]]).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019087.png" />-modular Frobenius groups; see [[#References|[a8]]]. Namely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019088.png" /> be a [[Field|field]] of positive characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019089.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019090.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019092.png" />-modular Frobenius group if it contains a non-trivial normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019093.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019094.png" /> is a splitting field for the [[Group algebra|group algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019095.png" /> and if one of the following (equivalent) statements holds:
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019096.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019097.png" /> has the property that the induced <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019098.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f12019099.png" /> is irreducible;
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A) every non-principal irreducible $K [ N ]$-module $V$ has the property that the induced $K [ G ]$-module $V ^ { G }$ is irreducible;
  
B) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190100.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190101.png" />-regular non-trivial element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190102.png" />. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190103.png" /> featuring in A)–B) is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190105.png" />-modular Frobenius kernel.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190106.png" />, the Frobenius kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190107.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190108.png" />-subgroup (cf. also [[Sylow subgroup|Sylow subgroup]]) of a Frobenius complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190109.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190110.png" /> is cyclic (cf. also [[Cyclic group|Cyclic group]]) when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190111.png" /> is odd, and cyclic or generalized quaternion if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190112.png" />. He also proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190113.png" /> 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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190114.png" />-modular Frobenius groups. Namely, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190115.png" />-modular Frobenius kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190116.png" /> is either solvable (cf. also [[Solvable group|Solvable group]]) or else <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190117.png" /> and any non–Abelian composition factor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190118.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190119.png" /> for some integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190120.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190121.png" /> is not a [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190122.png" />-group]]. Then:
+
Furthermore, assume that $N$ is not a [[P-group|$p$-group]]. Then:
  
any Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190123.png" />-subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190124.png" /> is cyclic whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190125.png" /> is relatively prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190126.png" />;
+
any Sylow $q$-subgroup of $G / N$ is cyclic whenever $q$ is relatively prime to $2 p$;
  
any Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190127.png" />-subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190128.png" /> is cyclic or generalized quaternion if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190129.png" /> is odd. On the other hand, any non-trivial finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190130.png" />-group (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190131.png" /> a prime number) is isomorphic to some quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190132.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190133.png" /> is a suitable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190134.png" />-modular Frobenius group with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190135.png" />-modular Frobenius kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190136.png" />. See also [[#References|[a8]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190138.png" />-transitive groups and finite simple groups (cf. also [[Transitive group|Transitive group]]; [[Simple finite group|Simple finite group]]).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190139.png" /> consist of the identity and those elements of the Frobenius group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190140.png" /> not occurring in any point stabilizer (cf. also [[Stabilizer|Stabilizer]]). Contrary to the finite case, it is now not always true that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190141.png" /> is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120190/f120190142.png" />. See [[#References|[a2]]] for examples.
+
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|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====
 
====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>
+
<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>

Revision as of 17:02, 1 July 2020

Suppose a 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), another is by viewing $G$ as a 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 [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 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, 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 [a7]).

This characterization led Kuisch, and later R.W. van der Waall, to the study of so-called $p$-modular Frobenius groups; see [a8]. Namely, let $K$ be a 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 $K [ N ]$ 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 [a9] showed that for a "classical" Frobenius group $G$, the Frobenius kernel $N$ is nilpotent (cf. also Nilpotent group), thereby solving a long-standing conjecture of W.S. Burnside. It was proved by H. Zassenhaus in 1939, [a10], that a Sylow $p$-subgroup (cf. also Sylow subgroup) of a Frobenius complement $H$ of $G$ is cyclic (cf. also 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), then it admits precisely one non-Abelian composition factor, namely the 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) 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. 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 [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; 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 [a2] for examples.

References

[a1] Yu.G. Berkovich, E.M. Zhmud, "Characters of finite groups" , Amer. Math. Soc. (1998/9)
[a2] J.D. Dixon, B. Mortimer, "Permutation groups" , GTM , 163 , Springer (1996)
[a3] G. Frobenius, "Ueber auflösbare Gruppen IV" Sitzungsber. Preuss. Akad. Wissenschaft. (1901) pp. 1216–1230
[a4] B. Huppert, "Endliche Gruppen" , I , Springer (1967)
[a5] B. Huppert, "Character theory of finite groups" , Experim. Math. , 25 , de Gruyter (1998)
[a6] I.M. Isaacs, "Character theory of finite groups" , Acad. Press (1976)
[a7] E.B. Kuisch, R.W. van der Waall, "Homogeneous character induction" J. Algebra , 156 (1993) pp. 395–406
[a8] E.B. Kuisch, R.W. van der Waall, "Modular Frobenius groups" Manuscripta Math. , 90 (1996) pp. 403–427
[a9] J.G. Thompson, "Finite groups with fixed point free automorphisms of prime order" Proc. Nat. Acad. Sci. USA , 45 (1959) pp. 578–581
[a10] H. Zassenhaus, "Ueber endliche Fastkörper" Abh. Math. Sem. Univ. Hamburg , 11 (1936) pp. 187–220
How to Cite This Entry:
Frobenius group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_group&oldid=19132
This article was adapted from an original article by R.W. van der Waall (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article