Solvable implies Fitting subgroup is self-centralizing
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
- Characteristicity is centralizer-closed
- Characteristicity is intersection-closed
- Members of the derived series of a group are characteristic subgroups (follows from the fact that characteristicity is commutator-closed)
- Characteristicity is quotient-transitive
- Characteristicity is transitive
Proof
Given: A solvable group . denotes the Fitting subgroup of , and denotes its centralizer in
To prove:
Proof: Let and . Note that every element of commutes with every element of , so , and in particular, is normal in .
Consider the derived series of . Since is solvable, so is , so its derived series terminates at the identity in finitely many steps. Let be the inverse image of the term just before the trivial subgroup in this derived series. Then, is a subgroup with the property that . But since commutes with every element of , it also commutes with every element of , so is trivial. Hence is nilpotent of class two.
We now show that is normal, through a series of observations:
- is a characteristic subgroup
- Since characteristicity is closed under taking centralizers, is also characteristic in
- Since characteristicity is closed under intersections, is characteristic in
- The quotient is a characteristic subgroup of , being a member of the derived series
- Hence, using the fact that characteristicity is quotient-transitive, is a characteristic subgroup of
- Since is already characteristic in , and using the fact that characteristicity is transitive, we see that is characteristic in
Thus, is a characteristic subgroup, hence a normal subgroup. So, is a nilpotent normal subgroup. Moreover, , but is not contained in , so cannot be contained in , contradicting the defining feature of as the subgroup generated by all nilpotent normal subgroups.
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 218, Theorem 1.3 (Section 6.1)