Frobenius Groups and the Normal Complement Theorem

Conjugation of subsets. Let $G$ be a group and let $S\subseteq G$ be a subset. For $g\in G$ write $gS{g}^{-1}:=\{gs{g}^{-1}:s\in S\},$ the image of $S$ under the inner automorphism $c_g$ of $G$. This is the convention of Conjugation $x\mapsto gx{g}^{-1}$ is an automorphism and Normal subgroup: invariance under conjugation: conjugation is written on the left, so that $gx{g}^{-1}$ is the conjugate of $x$ by $g$.

Frobenius complement. Let $G$ be a finite group. A subgroup $H\le G$ (Subgroup) with $\{1\}<H<G$ is a Frobenius complement of $G$ when

$H\cap gH{g}^{-1}=\{1\}\text{for every }g\in G\setminus H\mathrm{.}$

A group that possesses a Frobenius complement is a Frobenius group, and one then says that $G$ is a Frobenius group with complement $H$. Both subgroups $\{1\}$ and $G$ are excluded by the hypothesis $\{1\}<H<G$, so a Frobenius complement is a nontrivial proper subgroup.

• The condition is symmetric in $H$ and its conjugates. Since the condition is required only for $g\mathrm{\notin }H$ and is automatic for $g\in H$ (there $gH{g}^{-1}=H$, and $H\cap H=H\mathrm{\ne }\{1\}$ is not required to be trivial), one may equivalently require both $N_G(H)=H$ and $H\cap gH{g}^{-1}=\{1\}$ for all $g\in G$ with $gH{g}^{-1}\mathrm{\ne }H$. Replacing $g$ by $g^{-1}$ shows that $H\cap gH{g}^{-1}=\{1\}$ holds for all $g\mathrm{\notin }H$ if and only if $H^g\cap H=\{1\}$ holds for all $g\mathrm{\notin }H$, so the definition is not sensitive to using left rather than right conjugates.

• $H$ is its own normalizer. If $g\in N_G(H)$ (The normalizer $N_G(H)=\{g\in G:gH{g}^{-1}=H\}$ of a subgroup) then $gH{g}^{-1}=H$, hence $H\cap gH{g}^{-1}=H$; since $H\mathrm{\ne }\{1\}$ this forces $g\in H$. Thus for a Frobenius complement the normalizer is as small as possible, $N_G(H)=H$; this is used to count conjugates in Frobenius kernel cardinality.

• Terminology. The condition is a strong form of malnormality of $H$; the complement is not assumed to be normal, and the existence of the normal complement of Frobenius kernel theorem is the content of Frobenius' theorem, not part of this definition.

Let $G$ be a finite group and let $\{1\}<H<G$ be a subgroup. Then $H$ is a Frobenius complement of $G$ if and only if the left action of $G$ on the coset space $G\mathrm{/}H$ by left multiplication is transitive, is nonregular (that is, not free), and every nonidentity element of $G$ fixes at most one coset.

Let $G$ be a finite group and let $H$ be a Frobenius complement of $G$ (Frobenius complement and frobenius group). The candidate Frobenius kernel set, or simply the kernel set, attached to $H$ is

$N:=(G\setminus \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\bigcup }_{x\in G}xH{x}^{-1})\cup \{1\}\mathrm{.}$

Thus an element $g\in G$ lies in $N$ exactly when $g=1$ or $g$ lies in no conjugate $xH{x}^{-1}$ of $H$. Equivalently, by the fixed-point description of the coset action (Frobenius permutation action characterization), $N$ is the set of elements that either are the identity or fix no coset of $H$ in the left action of $G$ on $G\mathrm{/}H$; the identity is included by hand, because it lies in every conjugate of $H$.

Two cautions are part of the definition. First, $N$ is defined as a subset of $G$; no claim that $N$ is a subgroup is built into the notation, and the description "kernel" is provisional. Second, the set is invariant under conjugation: if $g\in N$ and $t\in G$ then $tg{t}^{-1}$ lies in no conjugate of $H$ whenever $g$ does, since $u(tg{t}^{-1})u^{-1}=(ut)g(ut{)}^{-1}$; this setwise invariance is not closure under products.

• Why "candidate". The counting argument of Frobenius kernel cardinality shows $\mathrm{\mid }N\mathrm{\mid }=[G:H]$, which is exactly the order a normal complement of $H$ would have to have. That argument alone produces no product in $N$; closure and normality are supplied only by the character-theoretic Frobenius kernel theorem, as recorded in Frobenius kernel closure is the content of the theorem ↗.

• Relation to the identity. Since $1\in xH{x}^{-1}$ for every $x$, the element $1$ must be added back by hand, and $N\cap H=\{1\}$: a nonidentity element of $H$ lies in the conjugate $1H{1}^{-1}=H$ and hence outside $N$. Both facts are used in Frobenius kernel cardinality.

Let $G$ be a finite Frobenius group with complement $H$ and let $N=(G\setminus {\bigcup }_{x\in G}xH{x}^{-1})\cup \{1\}$ be its kernel set. Then

$\mathrm{\mid }N\mathrm{\mid }=[G:H],N\cap H=\{1\}\mathrm{.}$

Let $G$ be a finite group, let $H\le G$ be a subgroup, and let $\theta \in \mathrm{c}\mathrm{f}(H)$ be a complex class function on $H$ (Class functions and the complex vector space $\mathrm{c}\mathrm{f}(G)$, Subgroup).

Induction. The induced class function ${\mathrm{Ind}}_H^G\theta \in \mathrm{c}\mathrm{f}(G)$ is defined by the Frobenius formula ${\mathrm{Ind}}_H^G\theta (g):=\frac{1}{\mathrm{\mid }H\mathrm{\mid }}{\sum }_{\begin{array}{c}x\in G\\ x^{-1}gx\in H\end{array}}\theta \left(x^{-1}gx\right),g\in G\mathrm{.}$ The sum is over the finite set of $x\in G$ with $x^{-1}gx\in H$, and each summand is a complex number, so the formula is well defined; the normalising factor $1\mathrm{/}\mathrm{\mid }H\mathrm{\mid }$ is the one used for honest induced characters in Frobenius' formula for the character of an induced representation.

Restriction. For $\psi \in \mathrm{c}\mathrm{f}(G)$ the restricted class function ${\mathrm{Res}}_H^G\psi \in \mathrm{c}\mathrm{f}(H)$ is the restriction $\psi {\mathrm{\mid }}_H$. It is a class function because the conjugating elements in the equation $f(hk{h}^{-1})=f(k)$ for $k,h\in H$ are also elements of $G$.

Elementary properties. The definition is arranged so that the following three statements hold; each is a direct computation from the displayed sum.

1. ${\mathrm{Ind}}_H^G\theta$ is a class function on $G$: for $t,g\in G$ the substitution $y=tx$ is a bijection from $\{x\in G:x^{-1}gx\in H\}$ onto $\{y\in G:y^{-1}(tg{t}^{-1})y\in H\}$, because $(tx{)}^{-1}(tg{t}^{-1})(tx)=x^{-1}gx$, and then ${\mathrm{Ind}}_H^G\theta (tg{t}^{-1})={\mathrm{Ind}}_H^G\theta (g)$.
2. Induction is $\mathbb{C}$-linear in $\theta$: for ${\theta }_1,{\theta }_2\in \mathrm{c}\mathrm{f}(H)$ and $\lambda \in \mathbb{C}$ one has ${\mathrm{Ind}}_H^G({\theta }_1+{\theta }_2)={\mathrm{Ind}}_H^G{\theta }_1+{\mathrm{Ind}}_H^G{\theta }_2$ and ${\mathrm{Ind}}_H^G(\lambda \theta )=\lambda {\mathrm{Ind}}_H^G\theta$, because both sides are the same finite sum of values of ${\theta }_1+{\theta }_2$, respectively $\lambda \theta$.
3. If $\theta =\chi$ is the character of a finite-dimensional complex representation of $H$, then ${\mathrm{Ind}}_H^G\chi$ as defined here is the honest induced character of The induced character ${\mathrm{Ind}}_H^G\chi$ of a complex character: this is exactly the content of the Frobenius formula Frobenius' formula for the character of an induced representation.

The map $\theta \mapsto {\mathrm{Ind}}_H^G\theta$ is thus a $\mathbb{C}$-linear map $\mathrm{c}\mathrm{f}(H)\to \mathrm{c}\mathrm{f}(G)$ extending the honest induction of characters; no claim of positivity, integrality or dependence only on a character is made for a general $\theta$.

• Why class functions are needed. The Frobenius kernel argument of this page applies induction to the virtual character $\theta =\phi -\phi (1)1_H$ of Frobenius character extension construction, which is not an honest character, and the library defines ${\mathrm{Ind}}_H^G\chi$ only for honest characters (The induced character ${\mathrm{Ind}}_H^G\chi$ of a complex character). The display above is the unique $\mathbb{C}$-linear extension of that construction, so Zero at identity induction restriction for a frobenius complement and Frobenius character extension construction may apply it to $\theta$. For an honest character $\chi$ of $H$ the two notions agree, by property 3.

• Value at the identity. At $g=1$ every $x\in G$ contributes to the defining sum, and $x^{-1}1x=1$, so ${\mathrm{Ind}}_H^G\theta (1)=\frac{1}{\mathrm{\mid }H\mathrm{\mid }}{\sum }_{x\in G}\theta (1)=[G:H]\theta (1)$. The normalising factor $1\mathrm{/}\mathrm{\mid }H\mathrm{\mid }$ is therefore what makes induction of the trivial character return $[G:H]$; for the virtual character $\theta =\phi -\phi (1)1_H$ of the kernel argument one has $\theta (1)=0$, so ${\mathrm{Ind}}_H^G\theta (1)=0$.

• Reciprocity. The $\mathbb{C}$-bilinear pairing between induction and restriction is recorded in Frobenius reciprocity for class functions.

Let $G$ be a finite group, let $H\le G$ be a subgroup, let $\theta \in \mathrm{c}\mathrm{f}(H)$ be a class function on $H$, and let $\psi \in \mathrm{c}\mathrm{f}(G)$ be a class function on $G$. Then

$⟨{\mathrm{Ind}}_H^G\theta ,\psi {⟩}_G=⟨\theta ,{\mathrm{Res}}_H^G\psi {⟩}_H\mathrm{.}$

Let $G$ be a finite Frobenius group with complement $H$, and let $\theta \in \mathrm{c}\mathrm{f}(H)$ be a complex class function on $H$ with $\theta (1)=0$. Then

${\mathrm{Res}}_H^G{\mathrm{Ind}}_H^G\theta =\theta \mathrm{.}$

Let $G$ be a finite Frobenius group with complement $H$. For a nontrivial irreducible complex character $\phi$ of $H$ put $\theta :=\phi -\phi (1)1_H$ and $\tilde{\phi }:={\mathrm{Ind}}_H^G\theta +\phi (1)1_G$, where $1_H$ and $1_G$ denote the constant function with value $1$ on $H$ and on $G$. Then

$\tilde{\phi }{\mid }_H=\phi ,\tilde{\phi }(1)=\phi (1),\tilde{\phi }(g)=\phi (1)\text{ for every }g\in G\text{ lying in no conjugate of }H\mathrm{.}$

Let $G$ be a finite Frobenius group with complement $H$ and let $\phi$ be a nontrivial irreducible complex character of $H$. Then $\tilde{\phi }={\mathrm{Ind}}_H^G\theta +\phi (1)1_G$ with $\theta =\phi -\phi (1)1_H$ is an irreducible complex character of $G$.