Group theory in SAGE

SAGE is an open source computer algebra system implemented using an object-oriented categorical framework, with methods for objects, methods for their elements, and methods for their morphisms. Currently, SAGE has the ability to deal with abelian groups, permutation groups, and matrix groups over a finite field. This paper will present an overview of the implementations of the group-theoretical algorithms in SAGE, with some examples. We conclude with some possible future directions. SAGE [?] is a general purpose computer algebra system started in 2005, built on top of existing open source packages, including GAP for group theory, Maxima for symbolic computation, Pari for number theory, and Singular for multivariate polynomial computations and commutative algebra. In design, SAGE uses the best of the ideas in Magma [?], Mathematica [?] and other systems, but uses the popular mainstream language Python as its interpreter. This paper will restrict itself to presenting an overview of the implementations of the group-theoretical algorithms in SAGE. According to the GAP website, each year between 50 and 100 papers are published which use GAP in an essential way. SAGE is far too young to have such an impressive research record. Still, it has been already used by several people for published research in coding theory, number theory, and modular forms, as well as being used in teaching both graduate and undergraduate math classes. Currently, SAGE has the ability to deal with abelian groups (finitely generated multiplicative abelian groups, groups of Dirichlet characters, and dual groups of finite abelian groups), permutation groups, matrix groups