# Finite Groups

GroupsGeneral group creation and standard group extension constructors.

[Groups](Groups/index-groups.md)Permutation GroupsPermutation group constructors, operations and homomorphisms.

[Permutation Groups](PermutationGroups/index-permutation-groups.md)Matrix Groups over General RingsGeneral linear group creation, homomorphisms and element operations.

[Matrix Groups over General Rings](MatrixGroupsOverGeneralRings/index-matrix-groups-over-general-rings.md)Matrix Groups over Finite FieldsElement order, Monte Carlo methods and Aschbacher reduction.

[Matrix Groups over Finite Fields](MatrixGroupsOverFiniteFields/index-matrix-groups-over-finite-fields.md)Matrix Groups over Infinite FieldsCongruence, finiteness and virtual membership tests.

[Matrix Groups over Infinite Fields](MatrixGroupsOverInfiniteFields/index-matrix-groups-over-infinite-fields.md)Matrix Groups over Q and ZInvariant forms, endomorphisms and related constructions.

[Matrix Groups over Q and Z](MatrixGroupsOverQAndZ/index-matrix-groups-over-qand-z.md)Finite Soluble GroupsPC-group definition, properties and homomorphisms.

[Finite Soluble Groups](FiniteSolubleGroups/index-finite-soluble-groups.md)Black-box GroupsBlack-box group creation, operations and element operations.

[Black-Box Groups](BlackBoxGroups/index-black-box-groups.md)Almost Simple GroupsDouble covers, standard constructions and group recognition.

[Almost Simple Groups](AlmostSimpleGroups/index-almost-simple-groups.md)Databases of GroupsSmall groups, simple groups and p-group databases.

[Databases of Groups](DatabasesOfGroups/index-databases-of-groups.md)Automorphism GroupsAutomorphism group creation, access and order.

[Automorphism Groups](AutomorphismGroups/index-automorphism-groups.md)Cohomology and ExtensionsCohomology groups, structural invariants and group extensions.

[Cohomology and Extensions](CohomologyAndExtensions/index-cohomology-and-extensions.md)Reductive GroupsReductive group creation, properties and operations.

[Reductive Groups](ReductiveGroups/index-reductive-groups.md)