Finite direct power-closed characteristic not implies fully invariant

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., finite direct power-closed characteristic subgroup) need not satisfy the second subgroup property (i.e., fully invariant subgroup)
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Statement

Verbal statement

It is possible to have a finite direct power-closed characteristic subgroup of a group that is not a fully invariant subgroup.

Facts used

  1. Center is finite direct power-closed characteristic
  2. Center not is fully invariant

Proof

Further information: direct product of S3 and Z2

The proof follows by piecing together facts (1) and (2).

An explicit example of (2), and hence of this result as well, is when the whole group G is the direct product of S3 and Z2 and H is the center of G.