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On an embedding property of generalized Carter subgroups

Citation

Cline, Edward T. (1966) On an embedding property of generalized Carter subgroups. Dissertation (Ph.D.), California Institute of Technology. http://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.))
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):
  • Hall, Marshall
Thesis Committee:
  • Unknown, Unknown
Defense Date:1 January 1966
Funders:
Funding AgencyGrant Number
CaltechUNSPECIFIED
NSFUNSPECIFIED
ARCS FoundationUNSPECIFIED
Record Number:CaltechTHESIS:09222015-083434916
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:09222015-083434916
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9163
Collection:CaltechTHESIS
Deposited By: Leslie Granillo
Deposited On:22 Sep 2015 19:48
Last Modified:22 Sep 2015 19:48

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