Fully invariant core

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This article defines a subgroup operator related to the subgroup property fully invariant 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

Definition with symbols

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

  1. It is the join of all fully invariant subgroups of G contained in H.
  2. It is the set of x \in H such that \sigma(x) \in H for all endomorphisms \sigma of G.
  3. It is the intersection of all the subgroups \sigma^{-1}(H) for all endomorphisms \sigma of G.