Normal not implies finite-pi-potentially 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., normal subgroup) need not satisfy the second subgroup property (i.e., finite-pi-potentially characteristic subgroup)
View all subgroup property non-implications | View all subgroup property implications


It is possible to have a finite group G and a normal subgroup H of G such that there is no finite group K containing G and satisfying both the following conditions:

Facts used

  1. Every finite p-group is a subgroup of a finite p-group that is not characteristic in any finite p-group properly containing it


By fact (1), we can definitely find a finite p-group H with the property that H is not a characteristic subgroup in any finite p-group properly containing it.

Let G be the direct product of H and a cyclic group of order p. H is thus a normal subgroup of G. However, for any finite p-group containing G, H is a proper subgroup of it and therefore not a characteristic subgroup of it. Hence, H cannot be made to be characteristic in any finite p-group K that contains G.