P residual of a finite group

Let $G$ be a finite group and let $p$ be a prime. A normal subgroup $N⊴G$ is $p$-cofinal when the quotient $G\mathrm{/}N$ is a finite $p$-group, that is, when $[G:N]$ is a power of $p$ (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 $\mathrm{\mid }G\mathrm{\mid }\mathrm{/}\mathrm{\mid }N\mathrm{\mid }$, A finite $p$-group has order $p^n$ for a prime $p$ and some $n\in \mathbb{N}$). The $p$-residual of $G$ is

$O^p(G):=\bigcap \{N⊴G:N\text{ is }p\text{-cofinal}\},$

the intersection of all $p$-cofinal normal subgroups of $G$. It is the unique smallest normal subgroup of $G$ whose quotient is a $p$-group: it is $p$-cofinal itself, and $O^p(G)\le N$ for every $p$-cofinal $N$.

Why the definition is well posed. The family $\mathcal{N}=\{N⊴G:G\mathrm{/}N$ is a $p$-group$\}$ is nonempty ($G\in \mathcal{N}$, since $G\mathrm{/}G$ is the trivial group, of order $p^0$), and it is finite, because every member is a subset of the finite set $G$ and a finite set has finitely many subsets (The cardinality $\mid A\mid$ of a finite set, $\mid \mathcal{P}(A)\mid =2^{\mid A\mid }$ for finite $A$). The intersection $O^p(G)$ is a subgroup of $G$ by The intersection of a nonempty family of subgroups of $G$ is a subgroup of $G$, and a normal subgroup because each $N$ is normal (Normal subgroup: invariance under conjugation); it remains to see that it is again $p$-cofinal, and least.

Finite intersections of $p$-cofinal subgroups are $p$-cofinal. Let $N_1,\dots ,N_r$ be $p$-cofinal. The diagonal map $\delta :G\to G\mathrm{/}{N}_1\times \cdots \times G\mathrm{/}{N}_r$, $\delta (g)=(g{N}_1,\dots ,g{N}_r)$, is a homomorphism of groups (Monoid homomorphism and group homomorphism); its kernel is $N_1\cap \cdots \cap N_r$, and its image is a subgroup of the direct product (First isomorphism theorem for groups: $G\mathrm{/}\ker f\cong \mathrm{im}f$, The image of a group homomorphism is a subgroup and its kernel is a normal subgroup). The direct product has order $\mathrm{\mid }G\mathrm{/}{N}_1\mathrm{\mid }\cdots \mathrm{\mid }G\mathrm{/}{N}_r\mathrm{\mid }$, a power of $p$ (For finite groups $G$ and $H$, $\mathrm{\mid }G\times H\mathrm{\mid }=\mathrm{\mid }G\mathrm{\mid }\mathrm{\mid }H\mathrm{\mid }$), hence is a finite $p$-group, and a subgroup of a finite $p$-group is a finite $p$-group (Every subgroup of a finite $p$-group has order a power of $p$). By the first isomorphism theorem $G\mathrm{/}(N_1\cap \cdots \cap N_r)\cong \mathrm{im}\delta$, so the intersection is $p$-cofinal.

The intersection of all of them is a member. Since $\mathcal{N}$ is finite, say $\mathcal{N}=\{N_1,\dots ,N_r\}$, the preceding paragraph shows that $O^p(G)=N_1\cap \cdots \cap N_r$ is $p$-cofinal; in particular $G\mathrm{/}{O}^p(G)$ is a $p$-group and $O^p(G)\le N$ for every $N\in \mathcal{N}$ by construction.

Smallest and unique. If $K⊴G$ has $G\mathrm{/}K$ a $p$-group, then $K\in \mathcal{N}$, so $O^p(G)\le K$; and $O^p(G)$ itself has $p$-group quotient. Hence $O^p(G)$ is the smallest normal subgroup of $G$ with $p$-group quotient, and it is the only one with that property, since two such subgroups contain each other.

The construction is the lower $p$-series counterpart of the $p^{\mathrm{\prime }}$-core $O_{p^{\mathrm{\prime }}}(G)$ of The p-prime core of a finite group: the $p^{\mathrm{\prime }}$-core is the largest normal $p^{\mathrm{\prime }}$-subgroup, while the $p$-residual is the smallest normal subgroup with $p$-group quotient. In particular $O^p(G)$ is the kernel of the natural map onto the largest $p$-group quotient of $G$, so every homomorphism from $G$ to a finite $p$-group factors through $G\mathrm{/}{O}^p(G)$ (The quotient group $G\mathrm{/}N$ and coset product $(gN)(hN)=ghN$).