Frobenius kernel set

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.