Characteristic core

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 defining ingredient::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 defining ingredient::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 defining ingredient::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 $$G$$.