# Intermediately characteristic not implies isomorph-containing in group of prime power order

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a group of prime power order. That is, it states that in a group of prime power order, every subgroup satisfying the first subgroup property (i.e., intermediately characteristic subgroup) neednotsatisfy the second subgroup property (i.e., isomorph-containing subgroup)

View all subgroup property non-implications | View all subgroup property implications

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 maximal subgroup of group of prime power order) neednotsatisfy the second subgroup property (i.e., isomorph-free subgroup of group of prime power order)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

Get more facts about characteristic maximal subgroup of group of prime power order|Get more facts about isomorph-free subgroup of group of prime power order

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property characteristic maximal subgroup of group of prime power order but not isomorph-free subgroup of group of prime power order|View examples of subgroups satisfying property characteristic maximal subgroup of group of prime power order and isomorph-free subgroup of group of prime power order

## Statement

For any prime number , there exists a finite -group and a subgroup of such that is an intermediately characteristic subgroup of but is not an isomorph-containing subgroup (and hence not an Isomorph-free subgroup (?), or Isomorph-free subgroup of group of prime power order (?)) of .

In fact, we can obtain an example where is a characteristic maximal subgroup of . Thus, a characteristic maximal subgroup of a group of prime power order need not be an Isomorph-free maximal subgroup of group of prime power order (?).

## Proof

`Further information: SmallGroup(16,4), subgroup structure of SmallGroup(16,4)`

For any , set to be SmallGroup(p^4,4) and to be the unique characteristic subgroup of order . is a direct product of a cyclic group of order and a cyclic group of order . All the other maximal subgroups of are also isomorphic to .

For instance, for , set to be SmallGroup(16,4) and to be the unique characteristic subgroup of order eight. Explicitly:

.

Both the other subgroups of order eight are isomorphic to , and all are isomorphic to the direct product of Z4 and Z2.