This volume contains the proceedings of the International Conference on Group Theory, Combinatorics and Computing held from October 3–8, 2012, in Boca Raton, Florida. The papers cover a number of areas in group theory and combinatorics. Topics include finite simple groups, groups acting on structured sets, varieties of algebras, classification of groups generated by 3-state automata over a 2-letter alphabet, new methods for construction of codes and designs, groups with constraints on the derived subgroups of its subgroups, graphs related to conjugacy classes in groups, and lexicographical configurations. Application of computer algebra programs is incorporated in several of the papers. This volume includes expository articles on finite coverings of loops, semigroups and groups, and on the application of algebraic structures in the theory of communications. This volume is a valuable resource for researchers and graduate students working in group theory and combinatorics. The articles provide excellent examples of the interplay between the two areas. Preface 8 List of Lectures 10 On the regularity of a graph related to conjugacy classes of groups: Small valencies 14 Introduction 14 1. F-groups 14 2. Graphs of diameter 3 16 3. Prime graphs with no complete vertices 16 4. Proof of the main result 17 References 20 Self-similar groups acting essentially freely on the boundary of the binary rooted tree 22 1. Introduction 22 2. Groups generated by automata and classification notations 25 3. Types of actions and main tools 30 4. Proof of the main theorem 35 5. Concluding remarks 58 References 59 Non-symmetric lexicographic configurations 62 1. Introduction 62 2. The notation 63 3. The first choice construction 64 4. The configuration E(k,r) 69 5. Examples 71 References 71 Algebraic structures for communications 72 1. Introduction 72 2. The ubiquitous Fourier matrix 74 3. Idempotent systems 77 4. Paraunitary matrices 78 5. Multiple antenna design 82 6. Coding theory 85 7. Crypto systems 87 8. Search engines, internet, threshold logic 88 9. Finally 89 References 89 Finite coverings: A journey through groups, loops, rings and semigroups 92 1. Introduction 92 2. Groups 94 3. Rings 95 4. Semigroups 96 5. Loops 97 6. Further results and some open problems 99 References 100 Decompositions of Kramer-Mesner matrices 102 1. Construction of t-designs 102 2. Adding Fixed Points 107 3. Direct Products: Blowing up Points 117 4. Homomorphisms of Group Actions 118 5. Prescribed Stabilizers 125 6. Backtrack Proposal 127 7. Results 129 References 132 Recent results on groups with few isomorphism classes of derived subgroups 134 1. Introduction 134 2. Some general results about the classes D_{n} 135 3. Some general results about the class D 137 4. Locally finite D2-groups 140 5. Soluble D2-groups 142 6. Other results on D2-groups 144 7. Locally finite D3-groups 144 References 148 Designs and codes from PSL2(q) 150 1. Introduction 150 2. Background and terminology 151 3. Method 1 154 4. Method 2 155 5. Designs and codes from PSL2(q) 156 References 160 Algebraic logic and logical geometry in arbitrary varieties of algebras 164 1. Logic for Universal Algebraic Geometry 164 2. Some results and problems 173 References 179 A uniform upper bound for the character degree sums and Gelfand-Graev-like characters for finite simple groups 182 1. Introduction 182 2. Preliminaries 185 3. Character degree sum for symmetric and alternating groups 186 4. Exceptional groups of Lie type 187 5. Linear groups 189 6. Unitary groups 191 7. Odd dimensional orthogonal groups 193 8. A uniform bound 194 9. The remaining classical groups and the proof of Theorem 1.3 197 References 199
