Distinct normal Sylow subgroups centralize one another

Given: The hypotheses and objects in the Statement.

Let $G$ be a finite group, let $p$ be prime, and write $\mathrm{\mid }G\mathrm{\mid }=p^{a}m$ with $a\in \mathbb{N}$ and $p\nmid m$. A subgroup $P\le G$ is a Sylow $p$-subgroup when $\mathrm{\mid }P\mathrm{\mid }=p^a$. Equivalently, its order is the largest power of $p$ dividing $\mathrm{\mid }G\mathrm{\mid }$. This is a property of a subgroup and does not presume that such a subgroup exists; existence is proved in thm-sylow-first-theorem. (Sylow $p$-subgroups of a finite group).

For subgroups $A,B\le G$, their subgroup commutator is $[A,B]=⟨[a,b]:a\in A,b\in B⟩,$ where $[a,b]=ab{a}^{-1}{b}^{-1}$ (def-commutator-and-commutator-subgroup, def-generated-subgroup). The lower central series is ${\gamma }_1(G)=G,{\gamma }_{r+1}(G)=[G,{\gamma }_r(G)](r\ge 1)\mathrm{.}$ Each ${\gamma }_r(G)$ is characteristic in $G$, and the series descends because $[G,N]\le N$ whenever $N⊴G$. (Subgroup commutators and the lower central series).

Let $G$ be a group and let $N\le G$ be a subgroup (def-subgroup). For $g\in G$, write $gN{g}^{-1}:=\{gn{g}^{-1}:n\in N\}\mathrm{.}$ The subgroup $N$ is normal in $G$ when $gN{g}^{-1}=N\text{for every }g\in G\mathrm{.}$ In that case write $N⊴G$. Equivalently, every inner conjugation of $G$ maps $N$ onto itself. The connection with equality of the left and right cosets of def-coset is proved in thm-normal-subgroup-characterisations. (Normal subgroup: invariance under conjugation).

Let $G$ be a finite group and $H\le G$. Then $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }\mathrm{.}$ Consequently, under the canonical embedding $\iota :\mathbb{N}\to \mathbb{Z}$, $\mathrm{\mid }H\mathrm{\mid }$ divides $\mathrm{\mid }G\mathrm{\mid }$. (Lagrange's theorem: $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }$ for every subgroup $H$ of a finite group $G$).