Date of Award
Fall 1982
Abstract
This thesis is an investigation of the interrelationships between a formation f, a finite solvable group G, and G(,f) the residual of f in G. This study is developed by introducing the f-subgroups. It is proven that the f-subgroups of G form a characteristic conjugacy class of CAR-subgroups of G. Moreover these subgroups generate G(,f). As a result, G is an element of the formation f if and only if an f-subgroup is equal to the identity subgroup.
It is established that an f-subgroup is a product of known subgroups of the f-residual. The covering and avoidance properties of f-subgroups are examined and the extent to which these properties characterize the f-subgroups is found.
Next, it is proven that an f-subgroup is a prefrattini subgroup when f is the formation of solvable nC-groups. Consequently, for the results obtained for f-subgroups corresponding results are valid for the prefrattini subgroups.
It is determined that a group G belongs to the formation f if and only if G has a series 1 = N(,0) (LESSTHEQ) N(,1) (LESSTHEQ) ... (LESSTHEQ) N(,n) = G such that N(,i+1)/N(,i) is a maximal nilpotent normal subgroup of G/N(,i) and the core of an f-subgroup of G/N(,i) is the identity subgroup for i = 0,1,...,n-1. This reduces to a corresponding result by G. Zacher when f is the solvable nC-groups.
Other CAR-subgroups that generate the f-residual are examined. An f-subgroup is proven to be the intersection of certain CAR-subgroups of the f-residual. A result by H. Bechtell is obtained as a corollary when f is the solvable nC-groups and an f-subgroup is a pefrattini subgroup.
A formation f of finite solvable groups is saturated if and only if for each group G every chief factor of the form G(,f)/K is complemented. In this work totally nonsaturated formations are defined as those formations in which G(,f)K is never complemented for any group G. It is proven that the structure of f-subgroups determine if a formation is saturated, totally nonsaturated, or neither of these two types of formations.
Document Type
Dissertation
Department or Program
Mathematics
Degree Name
Doctor of Philosophy
Recommended Citation
HOFMANN, MARK CHALLIS, "ON A CONJUGATE CLASS OF SUBGROUPS DETERMINED BY A FORMATION" (1982). Doctoral Dissertations. 1338.
https://scholars.unh.edu/dissertation/1338