The Department of Mathematics at Western Michigan University will present an algebra seminar on Mondays this fall semester.

**Day and time:** Mondays at 4 p.m.

**Place:** 6625 Everett Tower

The purpose of the Algebra Seminar is to reflect the research and scholarly interest of the Algebra Faculty: Clifton E. Ealy Jr – Quasi-groups, Groups, and related structures; Gene Freudenburg–Algebraic Geometry; Terrell Hodge – Algebraic groups and modular representation theory; John Martino–Group theory and classifying spaces of finite and compact groups; Annegret Paul–Representation of Lie groups; David Richter–Lie Algebras and matroids; and Jay Wood–coding theory over rings. The Algebra Seminar may be registered for as MATH 6930. The prerequisites for the Seminar as a course is graduate standing.

During the academic year 2017-18, our focus is Fusion Systems.

Other resources will be posted and/or shared separately, so please be sure to let Dr. Clifton E. Ealy Jr. know if you are interested in the seminar but were not able to attend the organizational meeting.

## upcoming events

## Past events

### Dec. 3

**The Frobenius Normal p-complement Theorem and Burnside's "nicht" p-nilpotent Theorem **presented by Clifton E. Ealy Jr., Ph.D., Department of Mathematics, Western Michigan University

**Abstract: **In this talk, I will, finally conclude the proof of Frobenius's Normal p-complement Theorem. I will also discuss Burnside's Theorem.

All are welcomed.

### Nov. 12

**The Frobenius p-complement Theorem and the Focal Subgroup Theorem **presented by Clifton E. Ealy Jr., Ph.D., Department of Mathematics, Western Michigan University

**Abstract: **In this talk, I will conclude the sketch of the proof of Frobenius's p-complement Theorem and, on the way, sketch the proof of the Focal Subgroup Theorem.

All are welcomed.

### Nov. 5

**The Frobenius p-complement Theorem and Burnside's criteria for not having a p-complement **presented by Clifton E. Ealy Jr., Ph.D., Department of Mathematics, Western Michigan University

**Abstract: **In this talk, I will sketch the proofs of Frobenius's p-complement Theorem and Burnside's criteria for not having a p-complement.

All are welcomed.

### Oct. 22

**Surreal numbers and the Sylow theorems for pro-finite groups **presented by Mohammad Shatnawi, Department of Mathematics, Western Michigan University

**Abstract: **In this talk, we will briefly consider the completion of the integers action on a pro-finite group, G. The main object of this talk will be to prove the Sylow Theorems for a pro-finite group G.

All are welcomed.

### Oct. 15

**pro-finite groups, the p-adic integers and the cyclic pro-finite groups **presented by Mohammad Shatnawi, Department of Mathematics, Western Michigan University

**Abstract: **In this talk, we will introduce the category of pro-finite groups. Our first example will be the p-adic integers, p a prime, an example of both a pro-finite ring and a pro-finite group. We will consider the completion of the integers as the inverse limit of (ℤ/mℤ |mεℕ); its Sylow subgroups; and its action on a pro-finite group. Finally, we will conclude with a discussion of all of the pro-finite cyclic groups up to isomorphism.

All are welcomed.

### Oct. 8

**pro-Lie groups, pro-finite groups & pro-finite graphs **presented by Clifton E. Ealy, Ph.D., Department of Mathematics, Western Michigan University

**Abstract:** In this talk, we will introduce the idea of the projective limit. First, following Karl Heinrich Hoffmann and Sydney A. Morris, we will consider how the category of pro-Lie groups relate to the category of locally compact groups. Next, time permitting, we will introduce pro-finite groups and pro-finite graphs, via Margulis's example of a family of expander graphs.

All are welcome.

### Sept. 18

**tdlc groups, tdc groups & tidy groups **presented by Clifton E. Ealy, Ph.D., Department of Mathematics, Western Michigan University

**Abstract:** In this talk, we will recall fusion systems of groups. Next, let G be a locally compact group and C the connected component of the identity then C is a characteristic subgroup of G and the sequence

1 → C → G → G/C (*)

is exact and G/C is a totally disconnected locally compact group, i.e., a tdlc group! We will look briefly at tdlc groups intoducing the tidy groups of G. Willis. To flesh out an introduction to tdlc groups go to:

All are welcome.

### Jan. 29

**The polar group of a real form of a complex variety **presented by Gene Freudenburg, Ph.D., Department of Mathematics, Western Michigan University

**Abstract:** Suppose that X is an algebraic variety over the real numbers R, and Y is an algebraic variety over the complex numbers C. Then X is called a real form of Y if Y is obtained from X by extension of the scalar field from R to C. For example, the punctured complex line C* has three distinct real forms, including the punctured real line R* (non-compact, non-connected) and the real 1-sphere S^{!} (compact, connected). Similarly, one defines real forms of integral domains over C, real forms of Lie algebras and Lie groups over C, and so on. The classification of real forms in these various contexts is an important endeavor with a rich history and a lot of current interest. We introduce polar groups as a tool for classifying real forms of affine varieties, and study its properties relative to properties of the underlying variety. Recently, Cassou-Nogues, Koras, Palka and Russell gave a description of all embeddings of C* in the complex plane C^{2}. Combining their description with the theory of polar groups, we show that the only polynomial embedding of **S ^{1}** in the plane R

^{2}is the standard one, given by x

^{2}+y

^{2}=1 for some system of coordinates (x,y). We conjecture that there is no polynomial embedding of the torus

**S**x

^{1}**S**in R

^{1}^{3}.

All are welcome.

### Dec. 11

**On the equivalence of the category of Sharply 2-transitive groups and the category of Near-domains **presented by Timothy Clark, Ph.D., Department of Mathematics, Adrian College

Abstract: In this talk I will demonstrate the equivalence of the category of Sharply 2-transitive groups and the category of Near-domains. Hence, if we prove a theorem in the category of Sharply 2-transitive groups we prove a corresponding theorem in the category of Near-domains and vice versa.

### Dec. 4

**Fusion in representation theory II **presented by Annegret Paul, Ph.D., Department of Mathematics, Western Michigan University

Abstract: The main topic of the Algebra Seminar at Western Michigan University this academic year is fusion systems. Fusion systems has its antecedents in the work of Burnside. We now venture into representation theory to see how fusion systems and representation theory, both ordinary and modular, interact. To each p-block of a finite group, in the group algebra, one may associate a fusion system. In this talk I will introduce the Brauer morphism, Brauer pairs, the defect group of a block b, and the fusion system of a block - time permitting.

### Nov. 27

**Fusion in representation theory I **presented by Annegret Paul, Ph.D., Department of Mathematics, Western Michigan University

Abstract: The main topic of the Algebra Seminar at Western Michigan University this academic year is fusion systems. Fusion systems has its antecedents in the work of Burnside. We now venture into representation theory to see how fusion systems and representation theory, both ordinary and modular, interact. To each p-block of a finite group, in the group algebra, one may associate a fusion system. In this first talk I will recall the classical Wedderburn theory, the role of idempotents in the group algebra, and introduce p-modular systems.

### Nov. 20

**Fusion in groups III **presented by Clifton E. Ealy, Jr., Ph.D., Department of Mathematics, Western Michigan University

Abstract: The main topic of the Algebra Seminar at Western Michigan University this academic year is fusion systems. Fusion systems has its antecedents in the work of Burnside. In this talk, following the David Craven lectures of Michaelmas Term, 2008, at Oxford, I will sketch a proof of the famous normal p-complement theorem of Georg Ferdinand Frobenius and discuss the non-existence theorem of Ronald Solomon. This theorem led to the exotic fusion systems Sol(q) for q, a power of a prime.

### Nov. 13

**Fusion in groups II **presented by Clifton E. Ealy, Jr., Ph.D., Department of Mathematics, Western Michigan University

Abstract: The main topic of the Algebra Seminar at Western Michigan University this academic year is fusion systems. Fusion systems has its antecedents in the work of Burnside. In this series of talks, I will give an overview of the rise of fusion and fusion systems in the study of groups.

### Nov. 6

**On Maschke's Theorem of Complete Reducibility **presented by Mohammad Shatnawi, Department of Mathematics, Western Michigan University

Abstract: In this talk we will introduce matrix representations of groups and character theory. We will then present a proof of Maschke's Theorem of Complete Reducibility. The presentation will be motivated by the treatment of group representations in Group Theory by Marshall Hall, Jr..

### Oct. 30

**A note on groups generated by involutions and sharply 2-transitive groups **presented by George Glauberman, Ph.D., Department of Mathematics, University of Chicago

Abstract: Let G be a finite or infinite group generated by a set C of elements of order two. We discuss conditions on C that yield that G is solvable or has a normal subgroup of index two consisting of elements of finite odd order, and we give an application to sharply doubly transitive permutation groups. This is joint work with A. Mann and Y. Segev.

### Oct. 23

**Fusion in groups I **presented by Clifton E. Ealy, Jr., Ph.D., Department of Mathematics, Western Michigan University

Abstract: The main topic of the Algebra Seminar at Western Michigan University this academic year is Fusion Systems. Fusion Systems has its antecedents in the work of Burnside; more recently in the works of Jonathan Alperin, George Glauberman, Don Higman, Ron Solomon and John G. Thompson. However, Fusion Systems present form was driven by the successful effort to prove the conjecture of John R. Martino and Stewart Priddy. In this series of talks I will give an overview of the rise of fusion and fusion systems in the study of finite groups.

### Oct. 16

**The beginnings of fusion; Burnside's normal p-complement theorem and Don Higman's focal subgroup theorem **presented by Mohammad Shatnawi, Department of Mathematics, Western Michigan University

Abstract: In this talk we will apply the transfer map to prove Burnside's normal p-complement theorem. Also, after proving some technical transfer lemmas, a sketch of Don Higman's focal subgroup theorem will be given. The presentation is motivated by the treatment of Die Verlagerung in Marshall Hall Jr's text Group Theory.

### Oct. 9

**Sharply 2-transitive groups of characteristic 2 **presented by Clifton E. Ealy, Jr., Ph.D., Department of Mathematics, Western Michigan University

Abstract: In this talk we will discuss sharply 2-transitive groups of characteristic 2 or equivalently fields, near-fields and near-domains of characteristic 2. So, first we will discuss infinite fields, near-fields and near-domains of characteristic 2. Then we will move to a single binary operation and consider infinite sharply 2-transitive groups. Finally, we will consider generic proper near-domains of characteristic 2.

### Oct. 2

**Die Verlagerung **presented by Mohammad Shatnawi, Department of Mathematics, Western Michigan University

Abstract: The transfer map is used in group theory, in group cohomology, in algebraic topology and in the study of fusion systems. In this talk we will introduce the transfer map. The presentation will be motivated by the treatment of Die Verlagerung in Marshall Hall Jr.'s text Group Theory.

### Sept. 25

**Sharply 2-transitive groups II **presented by Clifton E. Ealy, Jr., Ph.D., Department of Mathematics, Western Michigan University

Abstract: In this talk we will recall elementary properties of transitive and 2-transitive groups. Next, we will discuss the basic theorem (1939) of Reinhold Baer connecting group theory and loop theory. Finally, we will recall Karzel's construction (1968) of a near-domain from a sharply 2-trasitive group.

**Sept. 18**

**Sharply 2-transitive groups I **presented by Clifton E. Ealy, Jr., Ph.D., Department of Mathematics, Western Michigan University

Abstract: In this talk we will review sharply 2-transitive groups, near-fields and near-domains, as introduced in the Algebra Seminar during the academic year 2016-2017.

**Sept. 11**

Organizational Meeting presented by Clifton E. Ealy, Jr., Ph.D., Department of Mathematics, Western Michigan University

Abstract: This academic year the main thrust of the seminar will be Fusion Systems! But talks on other topics related to coding theory or algebraic geometry are welcomed. Historically, Fusion Systems arose in looking for conditions which tell us when a finite group, G, is not simple. For example, Burnside’s Theorem: If P ε Sylp(G) and P is a subgroup of Z(N G(P)), then G has a normal subgroup, H, such that G= H⋊P. Another example is Frobenius’s Normal p-complement Theorem: Let P ε Sylp(G). Then G= H⋊P if and only if N G(S) has a normal p-complement for every non-identity p-subgroup S of G. On the other hand, Frobenius’s Theorem on Frobenius groups: If G is transitive permutation group such that Gxy=1 whenever x≠y and H is the set of fixed point free elements of G with 1 included, then H is a subgroup of G and G= H⋊Gx., played an important role in the development of Fusion Systems. So, the Seminar maybe viewed as a continuation of last years Algebra Seminar. The seminar in the main will be based on David Cravens text: The Theory of Fusion Systems. But other references are Aschbacher and Oliver paper “Fusion Systems”, Bulletin of the AMS 10/2016; Aschbacher, Kessar, and Oliver’s text: Fusion Systems in Algebra and Topology; and the text: Finite Groups III, Chapter 10, Local Finite Group Theory by Huppert & Blackburn.