Order Functions

Unless the order is already known, each of the functions in this family will create a faithful permutation representation of the group of automorphisms in order to compute the order.

Order(A): GrpAuto -> RngIntElt

# A: GrpAuto -> RngIntElt

The order of the group of automorphisms \(A\), returned as an integer. If not already known, this function will create a permutation representation for \(A\).

FactoredOrder(A): GrpAuto -> [ <RngIntElt, RngIntElt> ]

The factored order of the group of automorphisms \(A\). If not already known, this function will create a permutation representation for \(A\).

OuterOrder(A): GrpAuto -> RngIntElt

The order of the outer automorphism group associated with the group of automorphisms \(A\).

Example: Autogp Order (ex-829895)

We create the non-soluble group \(G = PGL(2, 9)\) and examine the properties of its automorphism group.

> G := PGL(2, 9);
> A := AutomorphismGroup(G);
> A;
A group of automorphisms of GrpPerm: G, Degree 10, Order 2^4 * 3^2 * 5
Generators:
    Automorphism of GrpPerm: G, Degree 10, Order 2^4 * 3^2 * 5 which maps:
        (3, 5, 9, 6, 7, 4, 8, 10) |--> (1, 7, 3, 5, 4, 2, 10, 9)
        (1, 8, 2)(3, 4, 5)(6, 10, 7) |--> (1, 6, 8)(2, 7, 10)(3, 9, 5)
    Automorphism of GrpPerm: G, Degree 10, Order 2^4 * 3^2 * 5 which maps:
        (3, 5, 9, 6, 7, 4, 8, 10) |--> (1, 4, 6, 10, 7, 8, 5, 9)
        (1, 8, 2)(3, 4, 5)(6, 10, 7) |--> (1, 9, 10)(2, 6, 3)(4, 8, 7)
    Automorphism of GrpPerm: G, Degree 10, Order 2^4 * 3^2 * 5 which maps:
        (3, 5, 9, 6, 7, 4, 8, 10) |--> (3, 5, 9, 6, 7, 4, 8, 10)
        (1, 8, 2)(3, 4, 5)(6, 10, 7) |--> (1, 10, 2)(3, 4, 7)(5, 8, 9)
    Automorphism of GrpPerm: G, Degree 10, Order 2^4 * 3^2 * 5 which maps:
        (3, 5, 9, 6, 7, 4, 8, 10) |--> (1, 10, 3, 5, 2, 4, 7, 6)
        (1, 8, 2)(3, 4, 5)(6, 10, 7) |--> (1, 6, 2)(3, 7, 5)(4, 9, 8)
> #A;
1440
> FactoredOrder(A);
[ <2, 5>, <3, 2>, <5, 1> ]
> OuterOrder(A);
2
> InnerGenerators(A);
[
    Automorphism of GrpPerm: G, Degree 10, Order 2^4 * 3^2 * 5 which maps:
        (3, 5, 9, 6, 7, 4, 8, 10) |--> (1, 7, 3, 5, 4, 2, 10, 9)
        (1, 8, 2)(3, 4, 5)(6, 10, 7) |--> (1, 6, 8)(2, 7, 10)(3, 9, 5),
    Automorphism of GrpPerm: G, Degree 10, Order 2^4 * 3^2 * 5 which maps:
        (3, 5, 9, 6, 7, 4, 8, 10) |--> (1, 4, 6, 10, 7, 8, 5, 9)
        (1, 8, 2)(3, 4, 5)(6, 10, 7) |--> (1, 9, 10)(2, 6, 3)(4, 8, 7),
    Automorphism of GrpPerm: G, Degree 10, Order 2^4 * 3^2 * 5 which maps:
        (3, 5, 9, 6, 7, 4, 8, 10) |--> (3, 5, 9, 6, 7, 4, 8, 10)
        (1, 8, 2)(3, 4, 5)(6, 10, 7) |--> (1, 10, 2)(3, 4, 7)(5, 8, 9)
]
> CharacteristicSeries(A);
[
    Permutation group G acting on a set of cardinality 10
    Order = 720 = 2^4 * 3^2 * 5
        (3, 5, 9, 6, 7, 4, 8, 10)
        (1, 8, 2)(3, 4, 5)(6, 10, 7),
    Permutation group acting on a set of cardinality 10
    Order = 1
]

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