This memoir contains a complete classification of the finite irreducible 2-subgroups of $GL(4, {\mathbb C})$. Specifically, the author provides a parametrized list of representatives for the conjugacy classes of such groups, where each representative is defined by a generating set of monomial matrices. The problem is treated by a variety of techniques, including elementary character theory, a method for describing Hasse diagrams of submodule lattices, and calculation of 2-cohomology by means of the Lyndon-Hochschild-Serre spectral sequence. Related questions concerning isomorphism between the listed groups, and Schur indices of their defining characters, are also considered.

Features:

• A complete classification of a class of $p$-groups
• A first step towards extending presently available databases for use in proposed “soluble quotient algorithms”
• Groups presented explicitly; may be used to test conjectures or to serve generally as a resource in group-theoretic computations

Readership

Graduate students and research mathematicians interested in group theory and representation theory.