Commutator of a normal subgroup and a subgroup not implies normal

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Statement

It is possible to have a group L, a Normal subgroup (?) K of G, and a subgroup G of L, such that the Commutator of two subgroups (?) [K,G] is not a normal subgroup of L.

Related facts

Opposite facts

Other related facts

Proof

A generic example

Let G be a nontrivial perfect group. Consider the square K = G \times G (the external direct product of G with itself). Let L be the external semidirect product of K with the cyclic group of order two acting by the coordinate exchange automorphism (a,b) \mapsto (b,a).

In short L can also be defined as the external wreath product of G and a cyclic group of order two acting regularly.

Then, define G_1 = G \times \{ e \}, i.e., the first direct factor of K, and define G_2 = \{ e \} \times G, i.e., the second direct factor. Observe that:

  1. K is normal in L by construction.
  2. G_1 is a subgroup of L, but is not normal in L, because the coordinate exchange automorphism sends G_1 to G_2.
  3. G_1 is a direct factor of K, hence normal in K. Thus, [G_1,K] \le G_1.
  4. On the other hand, since G_1 \le K, [G_1,G_1] \le [G_1,K]. But since G_1 \cong G and G is perfect, we have G_1 = [G_1,G_1], so G_1 \le [G_1,K].
  5. Thus, by steps (3) and (4), we get G_1 = [G_1,K]. Step (2) tells us that G_1 is not normal in L. Thus, we have found a normal subgroup K and a subgroup G_1 such that the commutator [G_1,K] 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 G as the alternating group of degree five.

A more generic example

In the above example, we assumed that G was perfect. We can relax this slightly to assuming only that G is non-Abelian. In this more general case, we obtain that [G_1,K] = [G_1,G_1], which is a nontrivial subgroup of L contained inside G_1. This subgroup is not normal because the coordinate exchange automorphism sends it to a corresponding subgroup of G_2.

This more generic example allows us some nilpotent and solvable groups: