This semester the seminar will be dedicated to the study of groups and their relation to other branches of mathematics, in particular to geometry and graph theory.

Our textbook will be Groups and Their Graphs by Israel Grossman and Wilhelm Magnus (The Mathematical Association of America, New Mathematics Library). The book is unfortunately out of print, but some used copies are available on-line, and bound photocopies can be obtained as 640:196 text for $16 from Pequod (119 Somerset Street, New Brunswick, 732-214-8787).

The chapters assigned as reading below are chapters of this book. Don't be dismayed at being assigned two or even three chapters in one week: most chapters in this book are very short, in some cases under three pages.

We will also rely on seminar notes. Some of these are already posted at this site; further notes will be added by professors and students as we go along, so do check this site frequently.

There are some useful notes posted by students of this seminar. As an inspiring example from an earlier semester, look at the student webpage about Non-Euclidean Geometries by Aron Samkoff (Fall '03).

For bibliographical data, go to the excellent Scottish website of the University of St Andrews. You may want to start at the history page and browse, or go directly to the index of biographies.

January 18 and 25 (Prof. Carrington): Introduction to groups: basic definitions, examples familiar and unfamiliar.
Reading: Chapters 1, 2, and 3; and the following seminar notes: some basic algebraic structures, groups and fields, try these simple algebra problems

February 1 (Wei Chen and Tim Lou): Generators of a group, cyclic groups (finite and infinite), the dihedral groups, graph of a group, generators and relations.
Reading: Chapters 5, 6, and 7; seminar notes on cyclic groups.  Also read Tim's proof that the odd dihedral groups can be generated by two relations.

February 8 (Matt Calhoun and Daniel Greene): Isometries in Rⁿ and global positioning.
Reading: Seminar notes on isometries.

February 15 (Dunxu Hu, Sai Li, and Teresa Zhang): Modular arithmetic, subgroups, equivalence relations.
Reading: Chapters 3 and 8, and the following links.

February 22 (Joseph Shao): Symmetric and alternating groups.
Reading: Chapters 10, 13; a short summary, and the Appendix.

March 1 (Joonhee Lee and Martín Mosteiro): Isomorphism of groups, the group of symmetries of each Platonic solid.
Reading: The Appendix, and the website Symmetry groups of Platonic solids. See also the summary chart used by the speakers in class.

March 8 (Matt Calhoun and Tim Lou): The Euler-Descartes Formula for polyhedra and for plane graphs, and two of its consequences: (1) there are only five Platonic solids, (2) all plane maps can be colored with at most five colors.
Reading: Nineteen proofs; Platonic graphs; Planar graphs and plane graphs; Mathworld on the Four-Color Theorem; coloring maps on other surfaces.

March 22 (Grace Chen and Alex Sood): Subgroups, cosets, Lagrange's Theorem, normal subgroups.
Reading: Chapters 8 and 11; Prof. Lyons's 351 class notes.

March 29 (Wei Chen and Teresa Zhang): Finite groups of symmetry in R² and R³; another glorious appearance of the Platonic solids.
Reading: Topics in Geometry by John O'Connor, especially lectures 11 and 10.

April 5 (Peter Enny and Maggie Shinder): Some basic tools from linear algebra: orthogonal matrices. Proof that rotations in R³ fixing `the origin' form a group.
Reading: Topics in Geometry parts 2 and 4.   A finite group of isometries has a common fixed point.

April 12 (Sam Coskey, grad student): The Banach-Tarski Paradox -- an application of the group SO(3) to create madness.

April 19 (Grace Chen and Sushil Kumar): Groups and wallpaper designs.
Reading: Chapter 15; The 17 plane symmetry groups.

April 26 (Prof. Roe Goodman): Alice Through Looking Glass after Looking Glass; the Mathematics of Mirrors.
Reading:  The Mathematics of Mirrors and Kaleidoscopes.