Characteristic core

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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: largest characteristic subgroup inside, biggest characteristic subgroup inside, intersection of all automorphs

Symbol-free definition

The characteristic core of a subgroup is defined in the following equivalent ways:

  1. (Characteristic subgroup definition): As the subgroup generated by all characteristic subgroups of the whole group lying inside the subgroup; in other words, as the unique largest characteristic subgroup of the whole group contained in the given group.
  2. (Automorph-intersection definition): As the intersection of all automorphic subgroups to the given subgroup.

Definition with symbols

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

  1. (Characteristic subgroup definition): As the subgroup generated by all characteristic subgroups K of G such that K \le H; in other words, as the unique largest characteristic subgroup of the whole group contained in the given group.
  2. (Automorph-intersection definition): As the intersection of all subgroups of G of the form \sigma(H) where \sigma is an automorphism of G.