Normal p complement and p nilpotent group

Let $G$ be a finite group and let $p$ be a prime.

$p$-prime terminology. The $p$-element and $p^{\mathrm{\prime }}$-element language, for elements and for subgroups, is fixed once and for all in The p-prime core of a finite group and is used without further comment throughout the normal-complement material of this page.

Normal $p$-complement. A normal $p$-complement of $G$ is a normal subgroup $K⊴G$ (Normal subgroup: invariance under conjugation) such that

$p\nmid \mathrm{\mid }K\mathrm{\mid }\text{and}[G:K]\text{ is a power of }p\mathrm{.}$

Thus $K$ is a normal $p^{\mathrm{\prime }}$-subgroup whose index is a $p$-power. A group possessing a normal $p$-complement is called $p$-nilpotent.

Semidirect form. Let $P\in {\mathrm{Syl}}_p(G)$ (Sylow $p$-subgroups of a finite group). If $K$ is a normal $p$-complement then $[G:K]=\mathrm{\mid }P\mathrm{\mid }$ by Lagrange's theorem: $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }$ for every subgroup $H$ of a finite group $G$ and Sylow $p$-subgroups of a finite group, so $K\cap P=\{1\}$ and $KP=G$ with $K⊴G$; that is, $G=K⋊P$ (An internal semidirect product and a complement to a normal subgroup). Conversely, if $G=K⋊P$ for some $P\in {\mathrm{Syl}}_p(G)$, then $K$ is a normal $p^{\mathrm{\prime }}$-subgroup and $[G:K]=\mathrm{\mid }P\mathrm{\mid }$ is a $p$-power, so $K$ is a normal $p$-complement. In particular the definition is equivalent to the existence of a semidirect decomposition of $G$ with normal factor a $p^{\mathrm{\prime }}$-subgroup.

• Boundary cases. Both extremes are included. If $p\nmid \mathrm{\mid }G\mathrm{\mid }$ then $P=\{1\}$ and $K=G$ is a normal $p$-complement, so such a group is $p$-nilpotent vacuously; if $G$ is a $p$-group then $P=G$ and $K=\{1\}$ is a normal $p$-complement. In neither case is the complement required to be proper or nontrivial, in contrast to the Frobenius complement of Frobenius complement and frobenius group.

• Uniqueness. A normal $p$-complement, when it exists, is unique: if $K_1$ and $K_2$ are both of $p^{\mathrm{\prime }}$-order with $p$-power index, then $K_1{K}_2\mathrm{/}{K}_2\cong K_1\mathrm{/}(K_1\cap K_2)$ is both a subgroup of the $p$-group $G\mathrm{/}{K}_2$ and a quotient of the $p^{\mathrm{\prime }}$-group $K_1$. It is therefore trivial; hence $K_1\subseteq K_2$ and equality follows from $\mathrm{\mid }{K}_1\mathrm{\mid }=\mathrm{\mid }{K}_2\mathrm{\mid }=\mathrm{\mid }G\mathrm{\mid }\mathrm{/}\mathrm{\mid }P\mathrm{\mid }$. The identified complement is the $p^{\mathrm{\prime }}$-core $O_{p^{\mathrm{\prime }}}(G)$ of The p-prime core of a finite group, and the characterisations of Equivalent forms of having a normal p complement record further equivalent forms.

• Relation to the Frobenius condition. A finite Frobenius group with kernel $N$ and complement $H$ has $N$ as a normal $p$-complement whenever $p\nmid \mathrm{\mid }N\mathrm{\mid }$ and $\mathrm{\mid }H\mathrm{\mid }$ is a power of $p$: then $[G:N]=\mathrm{\mid }H\mathrm{\mid }$ is a $p$-power. In particular, Frobenius groups with a $p^{\mathrm{\prime }}$-kernel and a $p$-group complement are $p$-nilpotent in this sense; the link is drawn in Frobenius normal two complement for S_3 ↗.