Characteristic closure

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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This article defines a subgroup operator related to the subgroup property characteristic subgroup. By subgroup operator is meant an operator that takes as input a subgroup of a group and outputs a subgroup of the same group.

Definition

QUICK PHRASES: smallest characteristic subgroup containing, intersection of all characteristic subgroups containing, join of all automorphic subgroups

Symbol-free definition

The characteristic closure of a subgroup in a group can be defined in any of the following equivalent ways:

• As the intersection of all characteristic subgroups containing the given subgroup
• As the subgroup generated by all automorphic subgroups to the given subgroup
• As the set of all elements that can be written as products of finite length of elements from the subgroup and their automorphs

The characteristic closure of a subset is defined as the characteristic closure of the subgroup generated by that subset.

Definition with symbols

The characteristic closure of a subgroup $H$ in a group $G$ is defined in the following equivalent ways:

• As the intersection of all characteristic subgroups of $G$ containing $H$
• As the subgroup generated by all $\sigma(H)$ where $\sigma \in \operatorname{Aut}(G)$

The characteristic closure of a subset $A$ in $G$ is defined as the characteristic closure of the subgroup generated, i.e., of $\langle A \rangle$ in $G$.

Relation with other operators

Related operators

• Fully invariant closure
• Normal closure
• Characteristic core
• Normal core

Normal closure operator

The fact that characteristicity is the left transiter for normality has an interesting implication on the relation between characteristic closure and normal closure. Namely, given a subgroup $H$ of $G$, the characteristic closure of $H$ in $G$ is the smallest subgroup $L$ containing $H$ such that whenever $G \triangleleft K$, $L \triangleleft K$.

In other words, the normal closure of a subgroup may not remain a normal closure if we expand the bigger group. However,the characteristic closure remains normal even in an expanded bigger group (as long we we expand normally).

Facts

Characteristic closure of a minimal normal subgroup is a direct power of it

We can prove that for any minimal normal subgroup, the characteristic closure is a direct product of its automorphs. The idea of the proof is induction -- we build a collection of automorphs of the minimal normal subgroup. At each stage, we check if the automorphs generate the characteristic closure. If they do, then we have shown that the characteristic closure is a direct product of automorphs.

Otherwise, there is some automorph not in the direct product of automorphs so far. Since this automorph is again a minimal normal subgroup, it must intersect the direct product so far, trivially. Thus, we have got a bigger direct product.