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The representation theory of finite groups has seen rapid growth in recent years with the development of efficient algorithms and computer algebra systems. This is the first book to provide an introduction to the ordinary and modular representation theory of finite groups with special emphasis on the computational aspects of the subject. Evolving from courses taught at Aachen University, this well-paced text is ideal for graduate-level study. The authors provide over 200 exercises, both theoretical and computational, and include worked examples using the computer algebra system GAP. These make the abstract theory tangible and engage students in real hands-on work. GAP is freely available from www.gap-system.org and readers can download source code and solutions to selected exercises from the book's web page.
The representation theory of finite groups was developed around 1900 by Frobenius, Schur and Burnside. The theory was first concerned with representing groups by groups of matrices over the complex numbers or a field of characteristic zero. Representations over fields of prime characteristic, called “modular representations” (as opposed to “ordinary” ones), were considered somewhat later, and its theory began with fundamental papers by R. Brauer starting in 1935. Despite its age, the representation theory of finite groups is still developing vigorously and remains a very attractive area of research. In fact, the theory is notorious for its large number of longstanding open problems and challenging conjectures. The availability of computers, the development of algorithms and computer algebra systems within the last few decades have had some impact on representation theory, perhaps most noticeable by the appearance of the ATLAS of Finite Groups in 1985. Note that we refer to this as the ATLAS in the text.
The present book gives an introduction into representation theory of finite groups with some emphasis on the computational aspects of the subject. The book grew out of some sets of courses that the senior of the authors has given at Aachen University since the early 1990s. It was our experience that many students appreciated having many concrete examples illustrating the abstract theory.