Frobenius normal p complement theorem

Given: A finite group $G$, a prime $p$ and a Sylow $p$-subgroup $P\le G$.

(a) implies (b): if $G$ has a normal $p$-complement, then $N_G(Q)$ has a normal $p$-complement for every nontrivial $p$-subgroup $Q\le G$ (Normal p complements pass to subgroups and p local normalizers, P local normalizer for normal complement theory).

(b) implies (c): if every nontrivial $p$-local normalizer $N_G(Q)$, $1\mathrm{\ne }Q\le P$, has a normal $p$-complement, then $P$ controls fusion in $P$ with respect to $G$ (Local normal p complements force control of fusion, Control of fusion in a sylow p subgroup).

(c) implies (a): if $P$ controls fusion in $P$ with respect to $G$, then $P\cap O^p(G)=\{1\}$ (Fusion control forces trivial Sylow intersection with the p residual, P residual of a finite group).

$K:=O^p(G)$ is a normal subgroup of $G$, $G\mathrm{/}K$ is a finite $p$-group, and $G=PK$ (P residual of a finite group, P residual is generated by p prime elements and idempotent, Sylow $p$-subgroups of a finite group, A finite $p$-group has order $p^n$ for a prime $p$ and some $n\in \mathbb{N}$).

If $K⊴G$ then $PK$ is a subgroup of $G$ with $[G:K]=[PK:K]\cdot [G:PK]$ in the sense that $\mathrm{\mid }G\mathrm{\mid }=\mathrm{\mid }PK\mathrm{\mid }\cdot [G:PK]$, and $PK\mathrm{/}K\cong P\mathrm{/}(P\cap K)$; in particular, if $PK=G$ and $P\cap K=\{1\}$, then $\mathrm{\mid }G\mathrm{\mid }=\mathrm{\mid }P\mathrm{\mid }\mathrm{\mid }K\mathrm{\mid }$ and $[G:K]=\mathrm{\mid }P\mathrm{\mid }$ (Second isomorphism theorem for groups: $H\mathrm{/}(H\cap N)\cong HN\mathrm{/}N$, First isomorphism theorem for groups: $G\mathrm{/}\ker f\cong \mathrm{im}f$, If $H\le G$ and $N⊴G$, then $HN$ is a subgroup and $H\cap N⊴H$, If $[G:N]$ is finite then $\mathrm{\mid }G\mathrm{/}N\mathrm{\mid }=[G:N]$; for finite $G$ this equals $\mathrm{\mid }G\mathrm{\mid }\mathrm{/}\mathrm{\mid }N\mathrm{\mid }$, Lagrange's theorem: $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }$ for every subgroup $H$ of a finite group $G$, Normal subgroup: invariance under conjugation, Subgroup).

Order facts: $\mathrm{\mid }P\mathrm{\mid }$ is the exact power of $p$ dividing $\mathrm{\mid }G\mathrm{\mid }$, all Sylow $p$-subgroups of $G$ have this order, and $1=p^0$; if $S\le G$ then $\mathrm{\mid }S\mathrm{\mid }$ divides $\mathrm{\mid }G\mathrm{\mid }$ (Sylow $p$-subgroups of a finite group, Lagrange's theorem: $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }$ for every subgroup $H$ of a finite group $G$, A finite $p$-group has order $p^n$ for a prime $p$ and some $n\in \mathbb{N}$).