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.
","abstract_html":"<p>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.</p><p>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.</p><p>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.</p><p>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.</p><p>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.</p><p>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.</p>","abstract_has_math":false,"creators":["HOFMANN, MARK CHALLIS"],"institution":null,"degree_name":"Doctor of Philosophy","degree_level":"Dissertation","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1982,"date_issued":"1982-01-01T08:00:00Z","date_published":"1982-01-01T08:00:00Z","updated_at":"2026-07-24T05:23:15Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholars.unh.edu/dissertation/1338","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["HOFMANN, MARK CHALLIS"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholars.unh.edu/dissertation/1338"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["ON A CONJUGATE CLASS OF SUBGROUPS DETERMINED BY A FORMATION"]}]}],"canonical_facts":{"dc:creator":["HOFMANN, MARK CHALLIS"],"dc:description.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.
"],"dc:identifier":["https://scholars.unh.edu/dissertation/1338"],"dc:subject":["Mathematics"],"dc:title":["ON A CONJUGATE CLASS OF SUBGROUPS DETERMINED BY A FORMATION"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy"]},"updated_at":"2026-07-24T05:23:15Z"}