Let G be a finite group and A be a subgroup of Aut(G). In this work, we studied the influence of the index of fixed point subgroup of A in G on the structure of G.
When A is cyclic, we proved the following:
(1) [G,A] is solvable if this index is squarefree and the orders of G and A are coprime.
(2) G is solvable if the index of the centralizer of each x in H-G is squarefree where H denotes the semidirect product of G by A.
Moreover, for an arbitrary subgroup A of Aut(G) whose order is coprime to the order of G, we showed that when G is solvable, then the Fitting length f([G,A]) of [G,A] is bounded above by
the number of primes (counted with multiplicities) dividing the index of fixed point subgroup of A in G and this bound is best possible.
| Identifer | oai:union.ndltd.org:METU/oai:etd.lib.metu.edu.tr:http://etd.lib.metu.edu.tr/upload/12613522/index.pdf |
| Date | 01 August 2011 |
| Creators | Turkan, Erkan Murat |
| Contributors | Ercan, Gulin |
| Publisher | METU |
| Source Sets | Middle East Technical Univ. |
| Language | English |
| Detected Language | English |
| Type | Ph.D. Thesis |
| Format | text/pdf |
| Rights | To liberate the content for public access |
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