Characteristic not implies sub-(isomorph-normal characteristic) in finite

From Groupprops
Jump to: navigation, search
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a finite group, every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., sub-(isomorph-normal characteristic) subgroup)
View all subgroup property non-implications | View all subgroup property implications

Statement

We can have a finite group G and a characteristic subgroup H of G such that H is not sub-(isomorph-normal characteristic) in G. In other words, there is no ascending chain of subgroups from H to G such that each member is an isomorph-normal characteristic subgroup of its successor.

Related facts

Proof

The center of a non-abelian group of odd prime cube order

Further information: Prime-cube order group:U3p

Let p be an odd prime. Let G be the non-abelian group of order p^3 and exponent p. Let H be the center of G. Then, we have:

  • H is characteristic in G.
  • H is not sub-(isomorph-normal characteristic) in G: In fact, no proper subgroup of G containing H is isomorph-normal and characteristic. H itself is not isomorph-normal, because there are other cyclic groups of order p. Moreover, H is a maximal characteristic subgroup of G, so there is no other possibility.