0.1 core & socle

Characteristic group

It is mentioned in Characteristic Subgroup.

Def. If $G$ is a group, then $H$ is a characteristic subgroup of $G$ (written $H\mathrm{char}G$) if every automorphism of $G$ maps $H$ to itself.

Prop. If $K\mathrm{char}H$ and $H⊲G$ then $K⊲G$.

Proof. Consider the inner automorphisms of $G$ (automorphisms of the form $h\mapsto g^{-1}hg$ for some $g\in G$). These all preserve $H$, and so are automorphisms of $H$. But any automorphism of $H$ preserves $K$, so for any $g\in G$ and $k\in K$, $g^{-1}kg\in K$, i.e. $K⊲G$.

Def. A group is said to be characteristically simple if it has no proper nontrivial characteristic subgroups.

Prop. Any characteristically simple group is a product of isomorphic simple groups.

Proof. Take $T{⊲}_{\min }G$. Consider $M=⟨T^{\varphi }:\varphi \in \mathrm{Aut}(G)⟩$. Note that $T^{\varphi }$ is normal in $G$, so $$ M\cong T\times\cdots\times T.

Since $M$ is a characteristic group of $G$, $M=G$. ### Chief series **Def.** If $G$ is a group, then a *chief series* of $G$ is a finite collection of normal subgroups $N_i\lhd G$,

1=N_0\lhd N_1\lhd\cdots\lhd N_n=G

such that each quotient group $N_{i+1}/N_i$ for $i=1,\cdots n-1$ is a minimal normal subgroup of $G/N_i$, i.e., there does not exist any subgroup $A$ normal in $G$ such that $N_i<A<N_{i+1}$ for any $i$. The factor groups $N_{i+1}/N_i$ are called the *chief factors*. **Prop.** The chief factors are always characteristically simple. **Proof.** If there is non-trivial $L\space\mathrm{char}\space N_{i+1}/N_i$, then $L\lhd G/N_i$, which is impossible as $N_{i+1}/N_i$ is a minimal normal subgroup. **Cor.** A finite chief factor is a direct product of isomorphic simple groups. ### Core of group > It is mentioned [[Group Core|here]]. **Def.** For a group $G$, the *normal core* of a subgroup $H$ is the largest normal subgroup of $G$ that is contained in $H$, i.e. $\mathrm{Core}_G(H)=\cap_{g\in G}g^{-1}Hg$. > [!definition] > > For a prime $p$, the $p$-core of a finite group is defined to be its largest normal $p$-subgroup. It is the normal core of every Sylow p-subgroup of the group. The $p$-core of G is often denoted $O_p(G)$. ^slozsm **Def.** The *solvable radical* of $G$ is defined to be the largest solvable normal subgroup, and is denoted $O_\infty(G)$. **Lemma.** If $G=HK$ and $H$ is core-free in $G$, then $G$ acting on $[G:H]$ is faithful. **Proof.** If $Hg_ig=Hg_i$ for any $g_i\in G$, then $g^G\in H$ and so $g\in \cap_{g_i\in G} H^g_i=\mathrm{Core}_G(H)=\{1\}$. Thus $g=1$ and the action is faithful. ### Socle of group ![[Group Socle]]