P local normalizer for normal complement theory

Let $G$ be a finite group, $p$ a prime, and $P\in {\mathrm{Syl}}_p(G)$ a Sylow $p$-subgroup (Sylow $p$-subgroups of a finite group). On this page a nontrivial $p$-local normalizer of $G$ means a subgroup of the form

$N_G(Q)=\{g\in G:gQ{g}^{-1}=Q\},$

where $Q$ is a nontrivial subgroup of $P$, that is $1\mathrm{\ne }Q\le P$ (The normalizer $N_G(H)=\{g\in G:gH{g}^{-1}=H\}$ of a subgroup, Subgroup). The quantifier always excludes $Q=1$: since $N_G(1)=G$, admitting the trivial subgroup would make the local statements below vacuous or false.

Why normalizers suffice up to conjugacy. Every nontrivial $p$-subgroup $S$ of $G$ is contained in a Sylow $p$-subgroup of $G$, all of which are conjugate to $P$; so there is $g\in G$ with $S^g=gS{g}^{-1}\le P$ and $N_G(S^g)=N_G(S{)}^g$, by the conjugation automorphism Conjugation $x\mapsto gx{g}^{-1}$ is an automorphism (Sylow II: in a finite group every $p$-subgroup lies in a conjugate of any Sylow $p$-subgroup, and the Sylow $p$-subgroups form a single conjugacy class). Thus the family $\{N_G(Q):1\mathrm{\ne }Q\le P\}$ represents every normalizer of a nontrivial $p$-subgroup of $G$ up to conjugacy, and the claim "$N_G(Q)$ has a normal $p$-complement for every $1\mathrm{\ne }Q\le P$" is invariant under replacing $P$ by a conjugate Sylow subgroup. The normalizers are subgroups of $G$ by $C_G(x)$ and $N_G(H)$ are subgroups of $G$.

Centralizers are named separately. The broader convention in the literature calls both the normalizers $N_G(Q)$ and the centralizers $C_G(Q)$ of nontrivial $p$-subgroups $p$-local subgroups. The two theorems of this page that are stated locally — the inheritance lemma Normal p complements pass to subgroups and p local normalizers and Frobenius' normal $p$-complement theorem Frobenius normal p complement theorem — quantify over the normalizers $N_G(Q)$ only, so that is the meaning fixed here; centralizers (The centralizer $C_G(H)$ of a subgroup) are never silently included. A normalizer of a $p$-subgroup is itself a subgroup whose Sylow $p$-subgroups are again to be read with Sylow $p$-subgroups of a finite group and A finite $p$-group has order $p^n$ for a prime $p$ and some $n\in \mathbb{N}$.