Citation
Cline, Edward Thomas (1966) On an Embedding Property of Generalized Carter Subgroups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/S8GHXA56. https://resolver.caltech.edu/CaltechTHESIS:09222015083434916
Abstract
If E and F are saturated formations, we say that E is strongly contained in F if for any solvable group G with Esubgroup, E, and Fsubgroup, 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 = {GG/F(G) ϵT}, where T is a nonempty 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 Nsubgroups 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 nonempty formation of solvable groups, E = {GG/F(G) ϵT}, and E is strongly contained in F, then
(1) there is a formation V such that F = {GG/F(G) ϵV}.
(2) If for each prime p, we assume that T does not contain the class, S_{p’}, 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 = {GG/F(G) ϵS_{r’}}, then there are infinitely many formations which strongly contain R.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  (Mathematics)  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  1 January 1966  
Funders: 
 
Record Number:  CaltechTHESIS:09222015083434916  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:09222015083434916  
DOI:  10.7907/S8GHXA56  
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 
Thesis Files

PDF
 Final Version
See Usage Policy. 13MB 
Repository Staff Only: item control page