Commutator of a normal subgroup and a subgroup not implies normal
- Normality is commutator-closed: If both subgroups are normal, the commutator is also normal.
- Subgroup normalizes its commutator with any subset: In particular, both the subgroups normalize their commutator.
- Commutator of two subgroups is normal in join: In particular, if the subgroup generated by the two subgroups is the whole group, the commutator is normal.
- Commutator of a normal subgroup and a subset implies 2-subnormal: The commutator, even though not necessarily normal, must be a 2-subnormal subgroup.
- Commutator of a 2-subnormal subgroup and a subset implies 3-subnormal
- Commutator of a 3-subnormal subgroup and a finite subset implies subnormal
- Commutator of a 3-subnormal subgroup and a subset not implies subnormal
- Commutator of a 3-subnormal subgroup and a finite subset not implies 4-subnormal
A generic example
Let be a nontrivial perfect group. Consider the square (the external direct product of with itself). Let be the external semidirect product of with the cyclic group of order two acting by the coordinate exchange automorphism .
In short can also be defined as the external wreath product of and a cyclic group of order two acting regularly.
Then, define , i.e., the first direct factor of , and define , i.e., the second direct factor. Observe that:
- is normal in by construction.
- is a subgroup of , but is not normal in , because the coordinate exchange automorphism sends to .
- is a direct factor of , hence normal in . Thus, .
- On the other hand, since , . But since and is perfect, we have , so .
- Thus, by steps (3) and (4), we get . Step (2) tells us that is not normal in . Thus, we have found a normal subgroup and a subgroup such that the commutator is not normal.
Particular cases of this example
Further information: Alternating group:A5, A5 is simple, A5 is the simple non-Abelian group of smallest order
Any simple non-Abelian group is an example of a nontrivial perfect group, so the smallest particular case of this generic example is to set as the alternating group of degree five.
A more generic example
In the above example, we assumed that was perfect. We can relax this slightly to assuming only that is non-Abelian. In this more general case, we obtain that , which is a nontrivial subgroup of contained inside . This subgroup is not normal because the coordinate exchange automorphism sends it to a corresponding subgroup of .
This more generic example allows us some nilpotent and solvable groups: