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Groupprops β

Characteristic not implies potentially fully invariant

Statement

It is possible to have a subgroup H of a group G such that H is a characteristic subgroup of G but is not a potentially fully invariant subgroup of G, i.e., there is no group K containing G such that <H/math> is a [[fact about::fully invariant subgroup]] of <math>K.

Facts used

  1. Normal not implies potentially fully invariant: There exists an example of a group J and a normal subgroup H of J such that H is not fully invariant in any group K containing J.
  2. NPC theorem: This states that if H is a normal subgroup of J, there exists a group G containing J such that H is a characteristic subgroup of G.

Proof

Let J and H be a group-subgroup pair as given by fact (1). By fact (2), there exists a group G containing J such that H is a characteristic subgroup of G.

We want to show that there is no group K containing G such that H is fully invariant in K. The reason for this is: if such a group K exists, it would also contain J, so we'd have a group containing J in which H is fully invariant. This contradicts the choice of H and J as being examples for fact (1).