Explore: Special Aspects Of Infinite Or Finite Groups

algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects

considered has gradually expanded from finite permutation groups and special examples of matrix groups to abstract groups that may be specified through a presentation

(on a finite set of generators) or braid groups. Groups may also be infinitely presented, etc. An infinite group may be residually finite, meaning that every

goes beyond finite groups. The philosophy of cusp forms highlights the kinship of representation theoretic aspects of these types of groups with general

Lie groups as they are not finite-dimensional manifolds. Some totally disconnected groups, such as the Galois group of an infinite extension of fields

the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups, especially

"uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system

algebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their name, they do

invariant subspaces of V are the zero space and V itself. For certain types of Lie groups, namely compact and semisimple groups, every finite-dimensional representation

order of G. Every finite p-group is nilpotent. The remainder of this article deals with finite p-groups. For an example of an infinite abelian p-group, see