Characteristic not implies potentially fully invariant
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., characteristic subgroup) need not satisfy the second subgroup property (i.e., potentially fully invariant subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about characteristic subgroup|Get more facts about potentially fully invariant subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property characteristic subgroup but not potentially fully invariant subgroup|View examples of subgroups satisfying property characteristic subgroup and potentially fully invariant subgroup
- Normal not implies potentially fully invariant: There exists an example of a group and a normal subgroup of such that is not fully invariant in any group containing .
- NPC theorem: This states that if is a normal subgroup of , there exists a group containing such that is a characteristic subgroup of .
Let and be a group-subgroup pair as given by fact (1). By fact (2), there exists a group containing such that is a characteristic subgroup of .
We want to show that there is no group containing such that is fully invariant in . The reason for this is: if such a group exists, it would also contain , so we'd have a group containing in which is fully invariant. This contradicts the choice of and as being examples for fact (1).