Solvable implies Fitting subgroup is self-centralizing

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This article gives the statement, and possibly proof, of the fact that in any solvable group, the subgroup obtained by applying a given subgroup-defining function (i.e., Fitting subgroup) always satisfies a particular subgroup property (i.e., self-centralizing subgroup)
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This article gives the statement, and possibly proof, of a particular subgroup of kind of subgroup in a group being self-centralizing. In other words, the centralizer of the subgroup in the group is contained in the subgroup
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Statement

In a solvable group, the Fitting subgroup is self-centralizing: it contains its centralizer in the whole group.

Facts used

  1. Characteristicity is centralizer-closed
  2. Characteristicity is intersection-closed
  3. Members of the derived series of a group are characteristic subgroups (follows from the fact that characteristicity is commutator-closed)
  4. Characteristicity is quotient-transitive
  5. Characteristicity is transitive

Proof

Given: A solvable group G. F(G) denotes the Fitting subgroup of G, and C_G(F(G)) denotes its centralizer in G

To prove: C_G(F(G)) \le F(G)

Proof: Let C = C_G(F(G)) and H = C \cap F(G). Note that every element of H commutes with every element of C, so H \le Z(C), and in particular, H is normal in C.

Consider the derived series of C/H. Since G is solvable, so is C/H, so its derived series terminates at the identity in finitely many steps. Let B be the inverse image of the term just before the trivial subgroup in this derived series. Then, B \le G is a subgroup with the property that [B,B] \le H. But since H commutes with every element of C, it also commutes with every element of B, so [[B,B],B] is trivial. Hence B is nilpotent of class two.

We now show that B is normal, through a series of observations:

  1. F(G) is a characteristic subgroup
  2. Since characteristicity is closed under taking centralizers, C = C_G(F(G)) is also characteristic in G
  3. Since characteristicity is closed under intersections, H is characteristic in G
  4. The quotient B/H is a characteristic subgroup of C/H, being a member of the derived series
  5. Hence, using the fact that characteristicity is quotient-transitive, B is a characteristic subgroup of C
  6. Since C is already characteristic in G, and using the fact that characteristicity is transitive, we see that B is characteristic in G

Thus, B is a characteristic subgroup, hence a normal subgroup. So, B is a nilpotent normal subgroup. Moreover, B \le C, but B is not contained in H, so B cannot be contained in F(G), contradicting the defining feature of F(G) as the subgroup generated by all nilpotent normal subgroups.

References

Textbook references