ON A CONJUGATE CLASS OF SUBGROUPS DETERMINED BY A FORMATION
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.
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