Series-equivalent characteristic subgroups may be distinct

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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., characteristic subgroup) need not satisfy the second subgroup property (i.e., series-isomorph-free 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 series-isomorph-free subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property characteristic subgroup but not series-isomorph-free subgroup|View examples of subgroups satisfying property characteristic subgroup and series-isomorph-free subgroup

Statement

It is possible to have a group G, a characteristic subgroups H and K of G such that H and K are isomorphic groups and the quotient groups G/H and G/K are isomorphic groups. In other words, there can be distinct characteristic subgroups that are Series-equivalent subgroups (?).

In particular, a characteristic subgroup of G need not be a series-isomorph-free subgroup of G.

Related facts

Stronger facts

There are two slightly stronger facts that are true, either of which can be used to supply examples:

Proof

See the examples in the proof of either of the stronger facts.