117 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1983.This dissertation is a stu... more 117 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1983.This dissertation is a study of the relationship between the structure of a group and that of its automorphism group. We examine the characteristics of groups G whose automorphism group Aut G has specified properties. If H = Aut G for some group G, we call H an automorphism group. We will see that there are many classes of groups which are not automorphism groups, and many other classes which are automorphism groups of certain types of groups only.We begin with the construction, by homological techniques, of a type of central extension which is closely related to a stem extension. These extensions are used to describe all groups G such that Aut G is a finite semisimple group. Consequently, we are able to deduce necessary and sufficient conditions for a finite semisimple group to be an automorphism group. These conditions will imply that certain central extensions of a simple group by an abelian group are not automorphism groups. Similar arguments allow us to construct an infinite number of slightly more complicated semisimple groups which are the automorphism group of uncountably many infinite groups, but not of any finite group.Define a purely non-abelian group (p.n.a. group) to be a finite group whose composition factors are non-abelian. Purely non-abelian groups cannot be automorphism groups of infinite groups and we briefly consider the question of when such groups are automorphism groups. However, we are mainly concerned with characterizing all complete p.n.a. groups. This problem is intimately connected with the existence of self-normalizing subgroups of symmetric groups. We characterize those abstract groups which can occur as self-normalizing subgroups of a symmetric group and then construct examples of complete p.n.a. groups using such self-normalizing subgroups.Finally, we consider infinite torsion groups as automorphism groups. Let (omega)(H) be the set of all primes which divide the order of some torsion element of H. We show that if H is a countable torsion automorphism group with finite conjugacy classes and (omega)(H) finite, then H has finite exponent. This furnishes a large class of countable torsion groups which are not automorphism groups.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD
Bulletin of The London Mathematical Society, Jun 9, 2003
Let G be a finite group. The strong symmetric genus σ 0 (G) is the minimum genus of any Riemann s... more Let G be a finite group. The strong symmetric genus σ 0 (G) is the minimum genus of any Riemann surface on which G acts, preserving orientation. For any non-negative integer g, there is at least one group of strong symmetric genus g. For g = 2, one such group has the form Z k × D n for some k and n.
Let [Formula: see text] denote the set of positive integers that may appear as the real genus of ... more Let [Formula: see text] denote the set of positive integers that may appear as the real genus of a finite abelian group. We obtain a set of (simple) necessary conditions for an integer [Formula: see text] to belong to [Formula: see text]. We also prove that the real genus of an abelian group is not congruent to 3 modulo 4 and that the genus of an abelian group of odd order is a multiple of 4. Finally, we obtain upper and lower bounds for the density of the set [Formula: see text].
Let G be a finite group of odd order. The symmetric genus G is the minimum genus of any Riemann s... more Let G be a finite group of odd order. The symmetric genus G is the minimum genus of any Riemann surface on which G acts faithfully. Suppose G acts on a Riemann surface X of genus g ³ 2. If |G| > 8(g - 1), then |G| = K(g-1), where K is 15, 21/2, 9 or 33/4. We call these four types of groups LO1-groups through LO4-groups, respectively. We determine the characteristics of LO1-groups and LO2-groups and show that there are infinite families of each type. We also show that there are exactly ten odd order groups with genus between 2 and 26 inclusive. Finally, if G is an odd order group with symmetric genus of the form p+1 for an odd prime p or 2k + 1, for some positive integer k, then G is a metacyclic group with certain properties. We determine that, in the range between 26 and 200, most of the numbers of either of these forms are not the genus of an odd order group.
Let G be a finite group. The symmetric genus σ (G) is the minimum genus of any Riemann surface on... more Let G be a finite group. The symmetric genus σ (G) is the minimum genus of any Riemann surface on which G acts faithfully. We show that if G is a group of order 2 m that has symmetric genus congruent to 3 (mod 4), then either G has exponent 2 m-3 and a dihedral subgroup of index 4 or else the exponent of G is 2 m-2 . We then prove that there are at most 52 isomorphism types of these 2-groups; this bound is independent of the size of the 2-group G. A consequence of this bound is that almost all positive integers that are the symmetric genus of a 2-group are congruent to 1 (mod 4).
This paper looks at using the representation of small finite groups as groups of transformations ... more This paper looks at using the representation of small finite groups as groups of transformations of a compact surface of genus two or more to interest a general audience in the study of group theory. The ideas are presented in a very elementary way and the Portraits of Groups developed for previous Bridges conferences show interesting applications.
This paper represents a finite group with 32 elements as a group of transformations of a compact ... more This paper represents a finite group with 32 elements as a group of transformations of a compact surface of genus 5. In particular, we start with a designated pair of regions of this surface, and each region is labeled with the group element, which transforms the designated region into it. This gives a portrait of that finite group. These surfaces and the regions corresponding to the group elements are shown in this paper. William Burnside first gave a simple example of such a portrait in his 1911 book, "Theory of Groups of Finite Order". This paper is the third paper in a series which models groups as groups of transformations on a compact surface in the style of William Burnside.
This paper represents some small finite groups as groups of transformations of a compact surface ... more This paper represents some small finite groups as groups of transformations of a compact surface of small genus. In particular, we start with a designated pair of regions of this surface and each region is labeled with the group element, which transforms the designated region into it. This gives a portrait of that finite group. These surfaces and the regions corresponding to the group elements are shown in this paper. William Burnside first gave a simple example of such a portrait in his 1911 book, "Theory of Groups of Finite Order".
This course is the junior level interdisciplinary course of the Maryland Collaborative for Teache... more This course is the junior level interdisciplinary course of the Maryland Collaborative for Teacher Preparation program. The effectiveness of the course in teaching problem solving techniques and abstract mathematical ideas is documented. The students constructed their own knowledge from laboratory experiences involving digital logic circuits. They were subsequently challenged to abstract this knowledge and to find ways to solve progressively more difficult problems using these digital logic circuits. The mathematics of encoding and decoding information constituted the major mathematical content of the course. This course is shown to be effective in introducing prospective elementary and middle school teachers to abstract mathematical ideas and problem solving techniques.
The genesis of this paper is a talk that I gave to a non-mathematical audience in October 2007. T... more The genesis of this paper is a talk that I gave to a non-mathematical audience in October 2007. The audience was primarily faculty members in the College of Liberal Arts at Towson University. Many of the faculty confided to me that they were afraid of mathematics and not very good at it. The response to the talk was very positive and everyone felt that they got something out of it. A PowerPoint presentation of the talk is available at [5]. The talk should begin with a description of what we are going to do. Specifically, a group is a mathematical object like a number system. This part was held to a minimum. The concept of symmetry was discussed next. The members of the audience were given a set of cardboard equilateral triangles and pieces of paper with a slightly larger equilateral triangle drawn on it (as shown in Figure 1). The question was posed, how many symmetries of this equilateral triangle are there? The most common answers to this question by college students are three, fi...
Let S denote the set of positive integers that may appear as the strong symmetric genus of a fini... more Let S denote the set of positive integers that may appear as the strong symmetric genus of a finite abelian group. We obtain a set of (simple) necessary and sufficient conditions for an integer g to belong to S. We also prove that the set S has an asymptotic density and estimate its value.
This paper represents some small finite groups as groups of transformations of a compact surface ... more This paper represents some small finite groups as groups of transformations of a compact surface of small genus. In particular, we start with a designated pair of regions of this surface and each region is labeled with the group element, which transforms the designated region into it. This gives a portrait of that finite group. These surfaces and the regions corresponding to the group elements are shown in this paper. William Burnside first gave a simple example of such a portrait in his 1911 book, "Theory of Groups of Finite Order".
This paper looks at representing a group G of order 16 as a group of transformations of a compact... more This paper looks at representing a group G of order 16 as a group of transformations of a compact surface of genus three. This is the surface of smallest genus upon which G can be so represented. The action is realized as a quotient of a full quadrilateral group and so all regions have four edges. This group is also NOT the group of automorphisms of a regular map. There are significant differences between this portrait and previous ones.
Taylor & Francis makes every effort to ensure the accuracy of all the information (the "Content")... more Taylor & Francis makes every effort to ensure the accuracy of all the information (the "Content") contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.
Proceedings of the American Mathematical Society, 1986
We prove that if G G is a group such that Aut G G is a countably infinite torsion F C FC -group, ... more We prove that if G G is a group such that Aut G G is a countably infinite torsion F C FC -group, then Aut G G contains an infinite locally soluble, normal subgroup and hence a nontrivial abelian normal subgroup. It follows that a countably infinite subdirect product of nontrivial finite groups, of which only finitely many have nontrivial abelian normal subgroups, is not the automorphism group of any group.
Let A and B be finite groups and let S be the set of all extensions of A by B. A group G is calle... more Let A and B be finite groups and let S be the set of all extensions of A by B. A group G is called an extension cover of A B , if G contains all extensions in S as subgroups of G. A group G is called a minimal extension cover if G is an extension cover of minimal order. Let n = p e 1 1 • • • p e k k be the prime factorization of the odd number n and define n i = p e i i. The group D n 1 × • • • × D n k × Z 2 is the unique minimal extension cover of Z n Z 2. This article also constructs a minimal extension cover of Z 2 n Z 2. Some conjectures about minimal extension covers are examined as well.
117 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1983.This dissertation is a stu... more 117 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1983.This dissertation is a study of the relationship between the structure of a group and that of its automorphism group. We examine the characteristics of groups G whose automorphism group Aut G has specified properties. If H = Aut G for some group G, we call H an automorphism group. We will see that there are many classes of groups which are not automorphism groups, and many other classes which are automorphism groups of certain types of groups only.We begin with the construction, by homological techniques, of a type of central extension which is closely related to a stem extension. These extensions are used to describe all groups G such that Aut G is a finite semisimple group. Consequently, we are able to deduce necessary and sufficient conditions for a finite semisimple group to be an automorphism group. These conditions will imply that certain central extensions of a simple group by an abelian group are not automorphism groups. Similar arguments allow us to construct an infinite number of slightly more complicated semisimple groups which are the automorphism group of uncountably many infinite groups, but not of any finite group.Define a purely non-abelian group (p.n.a. group) to be a finite group whose composition factors are non-abelian. Purely non-abelian groups cannot be automorphism groups of infinite groups and we briefly consider the question of when such groups are automorphism groups. However, we are mainly concerned with characterizing all complete p.n.a. groups. This problem is intimately connected with the existence of self-normalizing subgroups of symmetric groups. We characterize those abstract groups which can occur as self-normalizing subgroups of a symmetric group and then construct examples of complete p.n.a. groups using such self-normalizing subgroups.Finally, we consider infinite torsion groups as automorphism groups. Let (omega)(H) be the set of all primes which divide the order of some torsion element of H. We show that if H is a countable torsion automorphism group with finite conjugacy classes and (omega)(H) finite, then H has finite exponent. This furnishes a large class of countable torsion groups which are not automorphism groups.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD
Bulletin of The London Mathematical Society, Jun 9, 2003
Let G be a finite group. The strong symmetric genus σ 0 (G) is the minimum genus of any Riemann s... more Let G be a finite group. The strong symmetric genus σ 0 (G) is the minimum genus of any Riemann surface on which G acts, preserving orientation. For any non-negative integer g, there is at least one group of strong symmetric genus g. For g = 2, one such group has the form Z k × D n for some k and n.
Let [Formula: see text] denote the set of positive integers that may appear as the real genus of ... more Let [Formula: see text] denote the set of positive integers that may appear as the real genus of a finite abelian group. We obtain a set of (simple) necessary conditions for an integer [Formula: see text] to belong to [Formula: see text]. We also prove that the real genus of an abelian group is not congruent to 3 modulo 4 and that the genus of an abelian group of odd order is a multiple of 4. Finally, we obtain upper and lower bounds for the density of the set [Formula: see text].
Let G be a finite group of odd order. The symmetric genus G is the minimum genus of any Riemann s... more Let G be a finite group of odd order. The symmetric genus G is the minimum genus of any Riemann surface on which G acts faithfully. Suppose G acts on a Riemann surface X of genus g ³ 2. If |G| > 8(g - 1), then |G| = K(g-1), where K is 15, 21/2, 9 or 33/4. We call these four types of groups LO1-groups through LO4-groups, respectively. We determine the characteristics of LO1-groups and LO2-groups and show that there are infinite families of each type. We also show that there are exactly ten odd order groups with genus between 2 and 26 inclusive. Finally, if G is an odd order group with symmetric genus of the form p+1 for an odd prime p or 2k + 1, for some positive integer k, then G is a metacyclic group with certain properties. We determine that, in the range between 26 and 200, most of the numbers of either of these forms are not the genus of an odd order group.
Let G be a finite group. The symmetric genus σ (G) is the minimum genus of any Riemann surface on... more Let G be a finite group. The symmetric genus σ (G) is the minimum genus of any Riemann surface on which G acts faithfully. We show that if G is a group of order 2 m that has symmetric genus congruent to 3 (mod 4), then either G has exponent 2 m-3 and a dihedral subgroup of index 4 or else the exponent of G is 2 m-2 . We then prove that there are at most 52 isomorphism types of these 2-groups; this bound is independent of the size of the 2-group G. A consequence of this bound is that almost all positive integers that are the symmetric genus of a 2-group are congruent to 1 (mod 4).
This paper looks at using the representation of small finite groups as groups of transformations ... more This paper looks at using the representation of small finite groups as groups of transformations of a compact surface of genus two or more to interest a general audience in the study of group theory. The ideas are presented in a very elementary way and the Portraits of Groups developed for previous Bridges conferences show interesting applications.
This paper represents a finite group with 32 elements as a group of transformations of a compact ... more This paper represents a finite group with 32 elements as a group of transformations of a compact surface of genus 5. In particular, we start with a designated pair of regions of this surface, and each region is labeled with the group element, which transforms the designated region into it. This gives a portrait of that finite group. These surfaces and the regions corresponding to the group elements are shown in this paper. William Burnside first gave a simple example of such a portrait in his 1911 book, "Theory of Groups of Finite Order". This paper is the third paper in a series which models groups as groups of transformations on a compact surface in the style of William Burnside.
This paper represents some small finite groups as groups of transformations of a compact surface ... more This paper represents some small finite groups as groups of transformations of a compact surface of small genus. In particular, we start with a designated pair of regions of this surface and each region is labeled with the group element, which transforms the designated region into it. This gives a portrait of that finite group. These surfaces and the regions corresponding to the group elements are shown in this paper. William Burnside first gave a simple example of such a portrait in his 1911 book, "Theory of Groups of Finite Order".
This course is the junior level interdisciplinary course of the Maryland Collaborative for Teache... more This course is the junior level interdisciplinary course of the Maryland Collaborative for Teacher Preparation program. The effectiveness of the course in teaching problem solving techniques and abstract mathematical ideas is documented. The students constructed their own knowledge from laboratory experiences involving digital logic circuits. They were subsequently challenged to abstract this knowledge and to find ways to solve progressively more difficult problems using these digital logic circuits. The mathematics of encoding and decoding information constituted the major mathematical content of the course. This course is shown to be effective in introducing prospective elementary and middle school teachers to abstract mathematical ideas and problem solving techniques.
The genesis of this paper is a talk that I gave to a non-mathematical audience in October 2007. T... more The genesis of this paper is a talk that I gave to a non-mathematical audience in October 2007. The audience was primarily faculty members in the College of Liberal Arts at Towson University. Many of the faculty confided to me that they were afraid of mathematics and not very good at it. The response to the talk was very positive and everyone felt that they got something out of it. A PowerPoint presentation of the talk is available at [5]. The talk should begin with a description of what we are going to do. Specifically, a group is a mathematical object like a number system. This part was held to a minimum. The concept of symmetry was discussed next. The members of the audience were given a set of cardboard equilateral triangles and pieces of paper with a slightly larger equilateral triangle drawn on it (as shown in Figure 1). The question was posed, how many symmetries of this equilateral triangle are there? The most common answers to this question by college students are three, fi...
Let S denote the set of positive integers that may appear as the strong symmetric genus of a fini... more Let S denote the set of positive integers that may appear as the strong symmetric genus of a finite abelian group. We obtain a set of (simple) necessary and sufficient conditions for an integer g to belong to S. We also prove that the set S has an asymptotic density and estimate its value.
This paper represents some small finite groups as groups of transformations of a compact surface ... more This paper represents some small finite groups as groups of transformations of a compact surface of small genus. In particular, we start with a designated pair of regions of this surface and each region is labeled with the group element, which transforms the designated region into it. This gives a portrait of that finite group. These surfaces and the regions corresponding to the group elements are shown in this paper. William Burnside first gave a simple example of such a portrait in his 1911 book, "Theory of Groups of Finite Order".
This paper looks at representing a group G of order 16 as a group of transformations of a compact... more This paper looks at representing a group G of order 16 as a group of transformations of a compact surface of genus three. This is the surface of smallest genus upon which G can be so represented. The action is realized as a quotient of a full quadrilateral group and so all regions have four edges. This group is also NOT the group of automorphisms of a regular map. There are significant differences between this portrait and previous ones.
Taylor & Francis makes every effort to ensure the accuracy of all the information (the "Content")... more Taylor & Francis makes every effort to ensure the accuracy of all the information (the "Content") contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.
Proceedings of the American Mathematical Society, 1986
We prove that if G G is a group such that Aut G G is a countably infinite torsion F C FC -group, ... more We prove that if G G is a group such that Aut G G is a countably infinite torsion F C FC -group, then Aut G G contains an infinite locally soluble, normal subgroup and hence a nontrivial abelian normal subgroup. It follows that a countably infinite subdirect product of nontrivial finite groups, of which only finitely many have nontrivial abelian normal subgroups, is not the automorphism group of any group.
Let A and B be finite groups and let S be the set of all extensions of A by B. A group G is calle... more Let A and B be finite groups and let S be the set of all extensions of A by B. A group G is called an extension cover of A B , if G contains all extensions in S as subgroups of G. A group G is called a minimal extension cover if G is an extension cover of minimal order. Let n = p e 1 1 • • • p e k k be the prime factorization of the odd number n and define n i = p e i i. The group D n 1 × • • • × D n k × Z 2 is the unique minimal extension cover of Z n Z 2. This article also constructs a minimal extension cover of Z 2 n Z 2. Some conjectures about minimal extension covers are examined as well.
Uploads
Papers by Jay Zimmerman