University of New Hampshire University of New Hampshire Scholars' Repository Doctoral Dissertations Student Scholarship Winter 1971 ON NORMAL FORMATIONS AND RELATED CLASSES OF GROUPS STEPHEN ALLAN BACON Follow this and additional works at: https://scholars.unh.edu/dissertation Recommended Citation BACON, STEPHEN ALLAN, "ON NORMAL FORMATIONS AND RELATED CLASSES OF GROUPS" (1971). Doctoral Dissertations. 941. https://scholars.unh.edu/dissertation/941 This Dissertation is brought to you for free and open access by the Student Scholarship at University of New Hampshire Scholars' Repository. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of University of New Hampshire Scholars' Repository. For more information, please contact [email protected]. 71-19,4-32 BACON, Stephen Allan, 1943- ON NORMAL FORMATIONS AND RELATED CLASSES OF GROUPS. University of New Hampshire, Ph.D., 1971 Mathematics University Microfilms, A XEROX Company, Ann Arbor, Michigan f THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED ON NORMAL FORMATIONS AND RELATED CLASSES OF GROUPS by / STEPHEN A. BACON .S., State University of New York at Albany, 1965 A THESIS Submitted to the University of New Hampshire In Partial Fulfillment of The Requirements for the Degree of Doctor of Philosophy Graduate School Department of Mathematics January, 1971 This thesis has been examined and approved. i l J U e M Thesis director, Homer P. Bechtell, Jr., Asso. Prof. of Mathematics of Mathematics Richard E. Johnson, P^pf. of Mathematics M ' ^ & A Z u )_________________ Edward H. Batho, Prof. of Mathematics ■\)./isLst v x ,• / /,c7v ,3^ ----------- ,____ ,_ ^/E4fic A. Nordgren, Atfso. Prof. of Mathematics ?; ^ 7 I It is a pleasure for the author to express appreciation and gratitude to Professor Homer F. Bechtell for his encouragement and guidance in the preparation of this thesis; Susan, my wife, for her patience and understanding; The National Aeronautics and Space Administration for its support for three years by means of a traineeship. TABLE OP CONTENTS ABSTRACT .............................................. V INTRODUCTION ................. 1 I. FUNDAMENTAL CONCEPTS ................................. 6 II. STRONG FORMATIONS ..................................... 16 1. Descending ^-systems ............................ 16 2. Ascending ^-systems ............................. 23 3. -series ........................................ 27 III. FORMATIONS OF FINITE GROUPS .......................... 33 IV. SATURATION ............................................ 49 V. APPLICATIONS AND EXAMPLES ............................ 55 1. Solvable K-groups ................................ 55 2. Hall (3%fr)-subgroups and -separable groups . 62 3. (3^71)-closed groups .............................. 70 VI. SUMMARY ............................................... 77 BIBLIOGRAPHY ......................................... 80 ABSTRACT ON NORMAL FORMATIONS AND RELATED CLASSES OF GROUPS r. by STEPHEN A. BACON This thesis is an investigation of the properties of several well-defined classes of groups and the rela­ tionships between them. The first of these is the normal formation. A formation is a nonempty class of groups which is closed with respect to epimorphic images and also has the property that for normal subgroups M and N of a group G, G / m £ ! > and G/ N G ^ imply G / M A N €J3r. A formation which satisfies the additional property that normal subgroups of members of & are also in is called a normal formation. Each formation gives rise to a normal forma­ tion ^ defined by & = {g | for all N ^ £ G, N e * }. Each normal formation in turn gives rise to a polyclass, i.e., a class of groups which is closed with respect to normal subgroups and epimorphic images and has the additional property that for each N ^ G, if N £ • © , G/N £ , then G € 4S. The polyclass associated with the normal formation 3^ is the class of all groups G which have a normal series of the form v G = An D A. ID ...DA = E u 1 n with Aj./Ai+|^ 9 . This is the "smallest" polyclass which contains the normal formation Qt. The third class of groups that is considered is called a Fitting formation. A Fitting formation SD is a normal formation which possesses the additional prop­ erty that for each group G and for each pair of normal subgroups N, M of G, N € 35, M £ ® imply NM € &. The classes of abelian, solvable and nilpotent groups are the obvious prototypes of these three general classes of groups. It is shown that many of the structural proper­ ties of the class of solvable groups are not dependent upon the fact that solvability is also a subgroup inher­ ited property. Also, many of the relationships between the derived series and the Fitting series of a solvable group carry over to the corresponding series determined by Fitting formations and their associated polyclasses. For a formation 9 each finite group G has a characteristic subgroup 9(G) with the property that G/N € 9 if and only if 9(G) S N. In this paper, the concept of formation is generalized to that of a strong formation so that infinite groups will also have this "commutator-type" subgroup. This generalization is made in such a way that the concepts of formation and strong formation are equivalent for finite groups. The notions of derived series and Fitting series for finite groups are generalized to ff’-derived systems and ^-Fitting vi systems with respect to a strong formation &. The properties of groups having either of these systems are examined fully. It is shown that the class of groups possessing an ^-Fitting system is a polyclass; for 3^- derived systems, this is not the case. For classes of finite groups, a number of conditions which are equiva­ lent to possessing a normal 3^-series are considered. It is shown that ^ if and only if 9 is a polyclass. A class 41 of groups is called saturated if and only if for each group G, G/J(G)€4I. implies G €. 5L • A number of conditions which imply saturation for the polyclass associated with a given normal formation are discussed. Among the conditions on ^ that imply sat­ uration for are the following: & is subgroup inherited; C £ Qfr for each prime p; or & is locally P defined. Also, negative answers are provided to the following questions: Is every saturated formation sub­ group inherited? Is every polyclass saturated? Three examples are discussed in detail in order to illustrate the theory and clarify the relationships between the classes of groups which have been considered. The first of these, the class of solvable K-groups, i.e., the class of solvable groups that split over each normal subgroup, is shown to be a normal formation that is not subgroup inherited. A second example is a generalization of the class of TT-separable groups, called (^,it)-separable groups. Results which generalize the theorems of P. Hall vii 1 on the existence and conjugacy of Hall 7T-subgroups in ff-separable groups are discussed. In addition, the class of groups having normal Hall 1t-subgroups which are members of a formation is considered, it is shown that the class of -separable groups is the polyclass associated with this class of groups. viii INTRODUCTION This thesis is an investigation of the properties of several well-defined classes of groups and the rela­ tionships between them. The first of these is the normal formation. A formation, as defined by Gaschutz £18}, is a nonempty class 3 s of groups which is closed with re­ spect to epimorphic images and also has the property that for normal subgroups M and N of a group G, G/M €. & and G/N € # imply G/mON A formation which sat­ isfies the additional property that normal subgroups of members of ^ are also in is called a normal forma­ tion. Unlike the earlier work of Gaschutz and Carter [9] on classes of conjugate subgroups in finite solvable groups, this thesis proceeds in a different direction in that the emphasis is on the properties of normal formations and classes of groups which they generate. Also, applications of formation theory to both non-solvable and infinite groups are considered. Each formation & gives rise to a normal formation defined by ^ = {g | for all N^£g, N & 9 Al­ though it is possible that & might reduce to the identity formation, the existence of groups in with common prop­ erties easily eliminates this possibility. Each normal formation in turn gives rise to a polyclass, i.e., a class 43 of groups which is closed with respect to normal sub­ groups and epimorphic images and has the additional property 1 that for each N A G, if N £ 45 , G/N € 35, then G € 35. The polyclass associated with the normal formation is the class of all groups G which have normal series of the form G = An 3 A. O . 3 A = E with A./A. _ €. u 1 n i i+l It is the "smallest" polyclass which contains the normal formation ^r. The obvious prototype of this type of sit­ uation is the polyclass of solvable groups which is asso­ ciated with the class of abelian groups. For finite groups there is another interesting class of groups "between" the abelian groups and the solv­ able groups called the nilpotent groups. The nilpotent groups serve as a prototype for the classes of groups called Fitting formations. A Fitting formation % is a normal formation which possesses the additional property that for each group G and for each pair of normal sub­ groups N, M of G, N G.'b, M G & imply NM €. It is shown that many of the structural properties of the class of solvable groups are not dependent upon the fact that solvability is also a subgroup inherited property. Also, many of the relationships between the derived series and the Fitting series for a solvable group carry over to the corresponding series determined by Fitting formations and their associated polyclasses. The classes of abelian, nilpotent, and solvable groups are each but one example of the more general classes of groups under consideration.