Normal not implies characteristic

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., normal subgroup) need not satisfy the second subgroup property (i.e., characteristic subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup but not characteristic subgroup|View examples of subgroups satisfying property normal subgroup and characteristic subgroup

Statement

It is possible to have a group $G$ and a normal subgroup $H$ of $G$ that is not a characteristic subgroup of $G$.

Related facts

Converse

Further information: Characteristic implies normal

The converse statement is indeed true. That is, every characteristic subgroup is normal.

Other related facts

• Normality is not transitive: A normal subgroup of a normal subgroup need not be normal.
• Characteristic of normal implies normal: A characteristic subgroup of a normal subgroup is normal.
• Left transiter of normal is characteristic: If $H$ is a subgroup of $K$ such that whenever $K$ is normal in $G$, $H$ is normal in $G$, then $H$ is characteristic in $K$.
• Direct factor not implies characteristic

Related group properties

There are some groups in which every normal subgroup is characteristic. Further information: Group in which every normal subgroup is characteristic

Proof

Example of a direct product

Let $H$ be any nontrivial group. Then consider $G = H \times H$, viz., the external direct product of $H$ with itself. The subgroups $H_1 := H \times \{ e \}$ and $H_2 := \{ e \} \times H$ are direct factors of $G$, and are hence both normal in $G$. Note also that they are distinct, since $H$ is nontrivial.

However, the exchange automorphism: $(x,y) \mapsto (y,x)$

exchanges the subgroups $H_1$ and $H_2$. Thus, neither $H_1$ nor $H_2$ is invariant under all the automorphisms, so neither is characteristic. Thus, $H_1$ and $H_2$ are both normal subgroups of $K$ that are not characteristic.

Note that this example also shows that direct factor does not imply characteristic subgroup.

Particular cases of this example

• When $H$ is the cyclic group of order two, $G$ is the Klein four-group. In particular, this gives a counterexample where the ambient group is an abelian group. More generally, we can start with any nontrivial abelian group $H$.

Initial examples

Here are some examples where the ambient group is quite important and easy to understand:

Group partSubgroup partQuotient part
Z2 in V4Klein four-groupCyclic group:Z2Cyclic group:Z2

Here are some examples where the ambient group is somewhat more complicated:

Group partSubgroup partQuotient part
Cyclic maximal subgroups of quaternion groupQuaternion groupCyclic group:Z4Cyclic group:Z2
Klein four-subgroups of dihedral group:D8Dihedral group:D8Klein four-groupCyclic group:Z2
Non-characteristic order two subgroups of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Cyclic group:Z4
Z4 in direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z4Cyclic group:Z2

Here are some examples where the ambient group is even more complicated:

Group partSubgroup partQuotient part
D8 in D16Dihedral group:D16Dihedral group:D8Cyclic group:Z2

GAP implementation

Implementation of the generic example

Before using this generic example, you need to define $H$ for GAP, choosing any nontrivial group (double semicolons have been used here to suppress GAP's output for the first three commands, which depends on the specific choice of $H$ -- you can use single semicolons instead).

gap> G := DirectProduct(H,H);;
gap> H1 := Image(Embedding(G,1));;
gap> H2 := Image(Embedding(G,2));;
gap> IsSubgroup(G,H1);
true
gap> IsSubgroup(G,H2);
true
gap> IsNormal(G,H1);
true
gap> IsNormal(G,H2);
true
gap> IsCharacteristicSubgroup(G,H1);
false
gap> IsCharacteristicSubgroup(G,H2);
false