Citation
Cline, Edward Thomas (1966) On an Embedding Property of Generalized Carter Subgroups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/S8GH-XA56. https://resolver.caltech.edu/CaltechTHESIS:09222015-083434916
Abstract
If E and F are saturated formations, we say that E is strongly contained in F if for any solvable group G with E-subgroup, E, and F-subgroup, F, some conjugate of E is contained in F. In this paper, we investigate the problem of finding the formations which strongly contain a fixed saturated formation E.
Our main results are restricted to formations, E, such that E = {G|G/F(G) ϵT}, where T is a non-empty formation of solvable groups, and F(G) is the Fitting subgroup of G. If T consists only of the identity, then E=N, the class of nilpotent groups, and for any solvable group, G, the N-subgroups of G are the Carter subgroups of G.
We give a characterization of strong containment which depends only on the formations E, and F. From this characterization, we prove:
If T is a non-empty formation of solvable groups, E = {G|G/F(G) ϵT}, and E is strongly contained in F, then
(1) there is a formation V such that F = {G|G/F(G) ϵV}.
(2) If for each prime p, we assume that T does not contain the class, Sp’, of all solvable p’-groups, then either E = F, or F contains all solvable groups.
This solves the problem for the Carter subgroups.
We prove the following result to show that the hypothesis of (2) is not redundant:
If R = {G|G/F(G) ϵSr’}, then there are infinitely many formations which strongly contain R.
Item Type: | Thesis (Dissertation (Ph.D.)) | ||||||||
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Subject Keywords: | (Mathematics) | ||||||||
Degree Grantor: | California Institute of Technology | ||||||||
Division: | Physics, Mathematics and Astronomy | ||||||||
Major Option: | Mathematics | ||||||||
Thesis Availability: | Public (worldwide access) | ||||||||
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Thesis Committee: |
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Defense Date: | 1 January 1966 | ||||||||
Funders: |
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Record Number: | CaltechTHESIS:09222015-083434916 | ||||||||
Persistent URL: | https://resolver.caltech.edu/CaltechTHESIS:09222015-083434916 | ||||||||
DOI: | 10.7907/S8GH-XA56 | ||||||||
Default Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||
ID Code: | 9163 | ||||||||
Collection: | CaltechTHESIS | ||||||||
Deposited By: | INVALID USER | ||||||||
Deposited On: | 22 Sep 2015 19:48 | ||||||||
Last Modified: | 27 Feb 2024 22:05 |
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