Frobenius automizer criterion for p nilpotence

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

(ii) implies (i): if the automizer condition (ii) holds, then by P automizer condition implies fusion control the Sylow $P$ controls fusion in $P$ with respect to $G$, and then by Frobenius normal p complement theorem $G$ has a normal $p$-complement (Control of fusion in a sylow p subgroup).

(i) implies (ii): suppose $G$ has a normal $p$-complement $K$, let $Q$ be a subgroup with $1\mathrm{\ne }Q\le P$, and put $N:=N_G(Q)$, $C:=C_G(Q)$ and $K_0:=K\cap N$. Then $K⊴G$ with $p\nmid \mathrm{\mid }K\mathrm{\mid }$ and $[G:K]$ a power of $p$, $K\cap P=\{1\}$, $K_0⊴N$, $Q⊴N$ and $Q\le N$ (Normal p complement and p nilpotent group, The normalizer $N_G(H)=\{g\in G:gH{g}^{-1}=H\}$ of a subgroup, Normal subgroup: invariance under conjugation, Subgroup, Sylow $p$-subgroups of a finite group).

Commutator inclusions used in step 1.2: if $A⊴N$ and $B\le N$, then $[A,B]\le A$, since for $a\in A$, $b\in B$ one has $ba{b}^{-1}\in A$ and hence $[a,b]=a(ba{b}^{-1}{)}^{-1}\in A$; and if $B⊴N$ then $[A,B]\le B$ likewise, since $ab{a}^{-1}\in B$; here $[A,B]=⟨[a,b]:a\in A,b\in B⟩$ with $[a,b]=ab{a}^{-1}{b}^{-1}$ (Commutators $[g,h]=gh{g}^{-1}{h}^{-1}$ and the commutator subgroup $[G,G]$, Subgroup commutators and the lower central series, Normal subgroup: invariance under conjugation, In a group $e^{-1}=e$, $(g^{-1}{)}^{-1}=g$ and $(gh{)}^{-1}=h^{-1}{g}^{-1}$, the order of the last product being essential, The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups).

Homomorphism and quotient facts: if $K⊴G$ then $g\mapsto gK$ is a homomorphism $G\to G\mathrm{/}K$ with kernel $K$, so $N\mathrm{/}{K}_0\cong NK\mathrm{/}K$ is isomorphic to a subgroup of $G\mathrm{/}K$; and for subgroups $K_0\le C\le N$ with both normal in $N$ the third isomorphism theorem gives $(N\mathrm{/}{K}_0)\mathrm{/}(C\mathrm{/}{K}_0)\cong N\mathrm{/}C$; subgroups and quotients of finite $p$-groups are finite $p$-groups (First isomorphism theorem for groups: $G\mathrm{/}\ker f\cong \mathrm{im}f$, Third isomorphism theorem for groups: $(G\mathrm{/}K)\mathrm{/}(N\mathrm{/}K)\cong G\mathrm{/}N$, The image of a group homomorphism is a subgroup and its kernel is a normal subgroup, Monoid homomorphism and group homomorphism, The quotient group $G\mathrm{/}N$ and coset product $(gN)(hN)=ghN$, 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 $G$, Every subgroup of a finite $p$-group has order a power of $p$, A finite $p$-group has order $p^n$ for a prime $p$ and some $n\in \mathbb{N}$).

If $P=\{1\}$ then $\mathrm{\mid }P\mathrm{\mid }=1=p^0$ is the exact power of $p$ dividing $\mathrm{\mid }G\mathrm{\mid }$, so $p\nmid \mathrm{\mid }G\mathrm{\mid }$ and $G$ itself is a normal $p$-complement of $G$; the condition (ii) is then vacuous (Sylow $p$-subgroups of a finite group, A finite $p$-group has order $p^n$ for a prime $p$ and some $n\in \mathbb{N}$, 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 p complement and p nilpotent group).