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 $N_i⊲G$,$1=N_0⊲N_1⊲\cdots ⊲N_n=G$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 $L\mathrm{char}{N}_{i+1}/N_i$, then $L⊲G/N_i$, 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 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 $O_p(G)$.

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 $H{g}_{i}g=H{g}_i$ for any $g_i\in G$, then $g^G\in H$ and so $g\in {\cap }_{g_i\in G}{H}_i^g={\mathrm{Core}}_G(H)=\{1\}$. Thus $g=1$ and the action is faithful.

Socle of group

Group Socle

Definition

The socle of a group is the subgroup generated by all its minimal normal subgroups.

Proposition

The socle of a finite group is a direct product of simple groups.

\begin{proof} Note that any different minimal normal subgroups intersects trivially, see here. Then by Minimal Normal Subgroup and Maximal Normal Subgroup the socle is a direct product of simple groups. \end{proof}

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