The Fitting and Frattini subgroups of $S_3$

Given: The hypotheses and objects in the Example.

For a finite group $G$, the Fitting subgroup is $F(G):={\prod }_{p\mid \mathrm{\mid }G\mathrm{\mid }}{O}_p(G),$ the product of its $p$-cores (def-p-core-of-a-finite-group). The factors are normal, so their finite product is a normal subgroup and does not depend on the order of multiplication. For the trivial group the product is empty and equals $1$. (The Fitting subgroup $F(G)={\prod }_p{O}_p(G)$ of a finite group).

For a finite group $G$, the Frattini subgroup is $\mathrm{\Phi }(G):=\bigcap \{M\le G:M\text{ is maximal proper}\}\mathrm{.}$ If $G=1$, the family is empty and its intersection inside $G$ is $G$ itself. Thus $\mathrm{\Phi }(1)=1$. (The Frattini subgroup $\mathrm{\Phi }(G)$ as the intersection of the maximal subgroups of a finite group).

For every finite group $G$, $F(G)$ is nilpotent and normal, and every normal nilpotent subgroup of $G$ is contained in $F(G)$. (The Fitting subgroup is nilpotent and is the largest normal nilpotent subgroup of a finite group).

Let $n\in \mathbb{N}$, so that $n=\{0,1,\dots ,n-1\}$ (def-natural-numbers). The symmetric group on $n$ letters is $S_n:=\mathrm{Sym}(n)=\mathrm{Sym}(\{0,1,\dots ,n-1\}),$ the group of all bijections of $n$ under composition (def-symmetric-group), with the composition convention. (The finite symmetric group $S_n$, one-line notation, and cycle notation).