Characteristic-isomorph-free not implies normal-isomorph-free 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. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic-isomorph-free subgroup) need not satisfy the second subgroup property (i.e., normal-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-isomorph-free subgroup|Get more facts about normal-isomorph-free subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property characteristic-isomorph-free subgroup but not normal-isomorph-free subgroup|View examples of subgroups satisfying property characteristic-isomorph-free subgroup and normal-isomorph-free subgroup

Statement

We may have a group with a characteristic subgroup such that there are no other characteristic subgroups isomorphic to it, but there are other normal subgroups isomorphic to it.

Proof

Example of an Abelian group

Further information: prime-cube order group:p2timesp, agemo subgroups of a group of prime power order, omega subgroups of a group of prime power order

Let p be any prime, and let G := \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p^2\mathbb{Z}. In other words, G is a direct product of cyclic groups of order p and p^2 respectively.

Consider the subgroup H := \operatorname{Agemo}^1(G): the set of all elements of G that can be written as p^{th} powers. This set is 0 \times p\mathbb{Z}/p^2\mathbb{Z}.

  • H is a characteristic subgroup: It is defined as the set of p^{th} powers, so it is clearly invariant under conjugation.
  • There is no other characteristic subgroup of G of order p: All the subgroups of order p in G lie inside \Omega_1(G) = \mathbb{Z}/p\mathbb{Z} \times p\mathbb{Z}/p^2\mathbb{Z}. Further, automorphisms of the form (a,b) \mapsto (a + b,b) permute all the other subgroups.
  • There are other normal subgroups of G of order p: In fact, there are p of them: the other p subgroups of \Omega_1(G).

Thus, H is characteristic-isomorph-free in G but is not normal-isomorph-free in G.