Frobenius complement and frobenius group

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.