Characteristic not implies isomorph-normal in finite group

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., isomorph-normal subgroup)
View all subgroup property non-implications | View all subgroup property implications

Statement

Statement with symbols

It is possible to have a group G and a subgroup H of G such that H is a characteristic subgroup of G but is not isomorph-normal in G: there exists a subgroup K of G isomorphic to H that is not normal in G.

Proof

Example of the dihedral group

Further information: dihedral group:D8

Let G be the dihedral group of order eight, given by:

G = \langle a,x \mid a^4 = x^2 = 1, xax = a^{-1} \rangle.

Let H be the center of G. H is a subgroup of order two generated by a^2.

  • H is characteristic.
  • H is not isomorph-normal: The subgroup \langle x \rangle of G is isomorphic to H, but is not normal in G, because conjugation by a sends it to the subgroup \langle a^2x \rangle.