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** this semester is to present an outline of the theory of complex semi-simple Lie algebras. The main goal is to reach the point where a novice can understand some structure theory of classical Lie groups and Lie algebras using Dynkin diagrams. **A secondary goal** is to understand the representation theory of complex semisimple Lie algebras. (These two goals are closely related.) Since this theory motivates or underlies many areas of contemporary research, students who plan to pursue a career in algebra or geometry are strongly urged to attend the Seminar.

The **prerequisites for the Seminar** are undergraduate abstract algebra and undergraduate linear algebra.

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 in September.

## upcoming events

### 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.

## PAST EVENTS

### 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.

All are welcomed

**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.

All are welcome