Matrix groups

These two topics are closely related. Here we consider some particular groups which arise most naturally as matrix groups or quotients of them, and special properties of matrix groups which are not shared by arbitrary groups. In representation theory, we consider what we learn about a group by considering all its homomorphisms to matrix groups. This article falls roughly into two parts. In the first part we discuss properties of specific matrix groups, especially the general linear group (consisting of all invertible matrices of given size over a given field) and the related “classical groups”. In the second part, we consider what we learn about a group if we know that it is a linear group. Most group theoretic terminology is standard and can be found in any textbook. A few terms we need are summarised in the next definition. Let X and Y be group-theoretic properties. We say that a group G is locally X if every finite subset of G is contained in a subgroup with property X; G is Xby-Y if G has a normal subgroup N such that N has X and G/N has Y; and G is poly-X if G has subgroups N0 = 1,N1, . . . ,Nr = G such that, for i = 0, . . . ,r− 1, Ni is normal in Ni+1 and the quotient has X . Thus a group is locally finite if and only if every finitely generated subgroup is finite; and a group is solvable if and only if it is poly-abelian.