Normal not implies characteristic in the collection of all groups satisfying a nontrivial finite direct product-closed group property
Suppose is the subcollection of the collection of all groups that satisfies a finite direct product-closed group property and contains at least one nontrivial group. Then, there exists a group and a subgroup of , such that is a Normal subgroup (?) of but not a Characteristic subgroup (?) of .
Note that this in particular applies to the situation where is a subvariety or a subquasivariety of the variety of groups.
Normal not implies characteristic in:
- the collection of finite groups.
- the collection of abelian groups.
- the collection of finite -groups for any prime number .
and all bigger varieties/quasivarieties.
Example of a direct product
Let be any nontrivial group in . Then consider , viz., the external direct product of with itself. The subgroups and are direct factors of , and are hence both normal in . Note also that they are distinct, since is nontrivial.
However, the exchange automorphism:
exchanges the subgroups and . Thus, neither nor is invariant under all the automorphisms, so neither is characteristic. Thus, and are both normal subgroups of that are not characteristic.
Implementation of the generic example
Before using this generic example, you need to define 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 -- you can use single semicolons instead).
gap> K := DirectProduct(G,G);; gap> G1 := Image(Embedding(K,1));; gap> G2 := Image(Embedding(K,2));; gap> IsSubgroup(K,G1); true gap> IsSubgroup(K,G2); true gap> IsNormal(K,G1); true gap> IsNormal(K,G2); true gap> IsCharacteristicSubgroup(K,G1); false gap> IsCharacteristicSubgroup(K,G2); false