The $p$-core $O_p(G)$ as the largest normal $p$-subgroup

For a finite group $G$ and a prime $p$, the $p$-core $O_p(G)$ is the subgroup generated by all normal $p$-subgroups of $G$. There are finitely many such subgroups. If $A$ and $B$ are two of them, then $AB$ is a subgroup by If $H\le G$ and $N⊴G$, then $HN$ is a subgroup and $H\cap N⊴H$ and is normal because $gAB{g}^{-1}=AB$ for every $g\in G$. For a fixed $ab\in AB$, the fibres of the multiplication map $A\times B\to AB$ are exactly the pairs $(ax,x^{-1}b)$ with $x\in A\cap B$, so

$$\mathrm{\mid }AB\mathrm{\mid }=\frac{\mathrm{\mid }A\mathrm{\mid }\mathrm{\mid }B\mathrm{\mid }}{\mathrm{\mid }A\cap B\mathrm{\mid }}\mathrm{.}$$

Lagrange's theorem Lagrange's theorem: $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }$ for every subgroup $H$ of a finite group $G$ makes $\mathrm{\mid }A\cap B\mathrm{\mid }$ a power of $p$, and therefore $AB$ is again a normal $p$-subgroup. Induction shows that the product of all normal $p$-subgroups is a normal $p$-subgroup. It contains every such subgroup, so it is the unique largest normal $p$-subgroup; this product is $O_p(G)$.