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A permutation group (G,X) is k-homogeneous if it is transitive on unordered k-subsets of X. The projective special linear group PSL(2,q) is 3-homogeneous if q=3 (mod 4).

An extension of a group H by a group N is a group G with a normal subgroup M such that M=N and G/M=H. This information can be encoded into a short exact sequence of groups ...

The term "homology group" usually means a singular homology group, which is an Abelian group which partially counts the number of holes in a topological space. In particular, ...

An algebraic group is a variety (or scheme) endowed with a group structure such that the group operations are morphisms of varieties (or schemes). The concept is similar to ...

A solvable group is a group having a normal series such that each normal factor is Abelian. The special case of a solvable finite group is a group whose composition indices ...

A simple group is a group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group. Simple groups ...

A Lie group is a group with the structure of a manifold. Therefore, discrete groups do not count. However, the most useful Lie groups are defined as subgroups of some matrix ...

The cyclic group C_(11) is unique group of group order 11. An example is the integers modulo 11 under addition (Z_(11)). No modulo multiplication group is isomorphic to ...

A set of generators (g_1,...,g_n) is a set of group elements such that possibly repeated application of the generators on themselves and each other is capable of producing ...

A group having an infinite number of elements. Some infinite groups, such as the integers or rationals, are not continuous groups.