# Matrix Groups over Infinite Fields

- [Overview](Overview.md)

- [Construction of Congruence Homomorphisms](congruence.md)

  - [`CongruenceImage(G : parameters): GrpMat → GrpMat, HomGrp, []`](congruence.md#function-congruenceimage-grpmat)

- [Testing Finiteness](finite-test.md)

  - [`IsFinite(G : parameters): GrpMat → BoolElt, RngIntElt`](finite-test.md#function-isfinite-grpmat)

  - [`IsomorphicCopy(G : parameters): GrpMat → BoolElt, GrpMat, HomGrp`](finite-test.md#function-isomorphiccopy-grpmat)

  - [`Order(G : parameters): GrpMat → RngIntElt`](finite-test.md#function-order-grpmat)

- [Deciding Virtual Properties of Linear Groups](virtual_tests.md)

  - [`IsSolubleByFinite(G : parameters): GrpMat → BoolElt`](virtual_tests.md#function-issolublebyfinite-grpmat)

  - [`IsPolycyclicByFinite(G : parameters): GrpMat → BoolElt`](virtual_tests.md#function-ispolycyclicbyfinite-grpmat)

  - [`IsNilpotentByFinite(G : parameters): GrpMat → BoolElt`](virtual_tests.md#function-isnilpotentbyfinite-grpmat)

  - [`IsAbelianByFinite(G : parameters): GrpMat → BoolElt`](virtual_tests.md#function-isabelianbyfinite-grpmat)

  - [`IsCentralByFinite(G : parameters): GrpMat → BoolElt`](virtual_tests.md#function-iscentralbyfinite-grpmat)

- [Hirsch Number and Pr{üfer Rank](hirsch_rank.md)

  - [`HasFiniteRank(G): GrpMat → BoolElt`](hirsch_rank.md#function-hasfiniterank-grpmat)

  - [`HirschNumber(G): GrpMat → RngIntElt`](hirsch_rank.md#function-hirschnumber-grpmat)

  - [`HasFiniteIndex(G, H): GrpMat, GrpMat → BoolElt`](hirsch_rank.md#function-hasfiniteindex-grpmat-grpmat)

  - [`PrueferRankBound(G): GrpMat → BoolElt`](hirsch_rank.md#function-prueferrankbound-grpmat)

- [Other Properties of Linear Groups](other-properties.md)

  - [`IsCompletelyReducible(G : parameters): GrpMat → BoolElt`](other-properties.md#function-iscompletelyreducible-grpmat)

  - [`CompletelyReduciblePart(G): GrpMat → GrpMat, GrpMatElt`](other-properties.md#function-completelyreduciblepart-grpmat)

  - [`IsUnipotent(G): GrpMat → BoolElt, GrpMatElt`](other-properties.md#function-isunipotent-grpmat)

  - [`IsNilpotent(G): GrpMat → BoolElt`](other-properties.md#function-isnilpotent-grpmat)

  - [`IsSoluble(G : parameters): GrpMat → BoolElt`](other-properties.md#function-issoluble-grpmat)

  - [`IsPolycyclic(G : parameters): GrpMat → BoolElt`](other-properties.md#function-ispolycyclic-grpmat)

  - [`HasFiniteOrder(g : parameters ): GrpMatElt → BoolElt, RngIntElt`](other-properties.md#function-hasfiniteorder-grpmatelt)

- [Other Functions for Nilpotent Matrix Groups](nilpotent_groups.md)

  - [`RecogniseAbelian(G): GrpMat → GrpGPC, Map, Map`](nilpotent_groups.md#function-recogniseabelian-grpmat)

  - [`SylowSystem(G : parameters): GrpMat[FldFin] → []`](nilpotent_groups.md#function-sylowsystem-grpmat-fldfin)

  - [`IsIrreducibleFiniteNilpotent(G : parameters): GrpMat → BoolElt, Any`](nilpotent_groups.md#function-isirreduciblefinitenilpotent-grpmat)

  - [`IsPrimitiveFiniteNilpotent(G : parameters): GrpMat → BoolElt, Any`](nilpotent_groups.md#function-isprimitivefinitenilpotent-grpmat)

- [Examples](examples.md)

  - [`Example: Is Finite Matrix Group FQ`](examples.md#example-ex-3405c0)

  - [`Example: Is Finite Matrix Group FF`](examples.md#example-ex-eadd93)

  - [`Example: Is Finite Matrix Group FF`](examples.md#example-ex-b89ce7)

  - [`Example: Is Finite Matrix Group FF`](examples.md#example-ex-83679b)

  - [`Example: Is Finite Matrix Group FF`](examples.md#example-ex-cd9712)

  - [`Example: Is Finite Matrix Group`](examples.md#example-ex-d1ef57)

  - [`Example: Is Finite Matrix Group F`](examples.md#example-ex-e6bb73)

  - [`Example: Is Finite Matrix Group F`](examples.md#example-ex-b2fc02)

  - [`Example: Is Finite Matrix Group F`](examples.md#example-ex-65cc00)

  - [`Example: Is Finite Matrix Group F`](examples.md#example-ex-b02d39)

  - [`Example: Is Nilpotent Matrix Group F`](examples.md#example-ex-1259d9)

  - [`Example: Is Nilpotent Matrix Group F`](examples.md#example-ex-a460de)

  - [`Example: Is Nilpotent Matrix Group F`](examples.md#example-ex-644306)

  - [`Example: Is Nilpotent Matrix Group F`](examples.md#example-ex-8207b7)