Editing Group (mathematics) (section)

=== General linear group and representation theory ===
{{Main|General linear group|Representation theory|Character theory}}
[[Image:Matrix multiplication.svg|right|thumb|250px|Two [[vector (mathematics)|vectors]] (the left illustration) multiplied by matrices (the middle and right illustrations). The middle illustration represents a clockwise rotation by 90°, while the right-most one stretches the {{tmath|1= x }}-coordinate by factor 2.|alt=Two vectors have the same length and span a 90° angle. Furthermore, they are rotated by 90° degrees, then one vector is stretched to twice its length.]]
[[Matrix group]]s consist of [[Matrix (mathematics)|matrices]] together with [[matrix multiplication]]. The ''general linear group'' <math>\mathrm {GL}(n, \R)</math> consists of all [[invertible matrix|invertible]] {{tmath|1= n }}-by-{{tmath|1= n }} matrices with real entries.{{sfn|Lay|2003}} Its subgroups are referred to as ''matrix groups'' or ''[[linear group]]s''. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the [[special orthogonal group]] {{tmath|1= \mathrm{SO}(n) }}. It describes all possible rotations in <math>n</math> dimensions. [[Rotation matrix|Rotation matrices]] in this group are used in [[computer graphics]].{{sfn|Kuipers|1999}}

''Representation theory'' is both an application of the group concept and important for a deeper understanding of groups.{{sfn|Fulton|Harris|1991}}{{sfn|Serre|1977}} It studies the group by its group actions on other spaces. A broad class of [[group representation]]s are linear representations in which the group acts on a vector space, such as the three-dimensional [[Euclidean space]] {{tmath|1= \R^3 }}. A representation of a group <math>G</math> on an <math>n</math>-[[dimension]]al real vector space is simply a group homomorphism
<math>\rho : G \to \mathrm {GL}(n, \R)</math>
from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.{{efn|This was crucial to the classification of finite simple groups, for example. See {{harvnb|Aschbacher|2004}}.}}

A group action gives further means to study the object being acted on.{{efn|See, for example, [[Schur's Lemma]] for the impact of a group action on [[simple module]]s. A more involved example is the action of an [[absolute Galois group]] on [[étale cohomology]].}} On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and [[topological group]]s, especially (locally) [[compact group]]s.{{sfn|Fulton|Harris|1991}}{{sfn|Rudin|1990}}