Representation theory of groups

group homomorphism into the general linear group over a vector space

operation of the elements of a group as transformations or automorphisms (mathematics)

class function associated to any finite-dimensional group representation, defined as the trace of the representation matrices

free module and at the same time a ring

homomorphism G → PGL(V) from a group G to a projective linear group PGL(V) over a vector space V

thumb|The torus can be made an abelian group isomorphic to the product of the [[circle group. This abelian group is a Klein four-group-module, where the group acts by reflection in each of the coordinate directions (here depicted by red and blue arrows intersecting at the identity element).]]

Generalization of Lie groups

group of group homomorphisms from a group into the multiplication group of nonzero elements of a field

function on a group