Subsections

• Characteristic Subgroups and Subgroup Series
• Normal Structure of a Primitive Group
• The Abstract Structure of a Group

Center(G) : GrpPerm -> GrpPerm

Construct the centre of the group G.

Hypercenter(G) : GrpPerm -> GrpPerm

Construct the hypercentre of the group G (the stationary term of the upper central series).

The derived length of G. If G is non-soluble, the function returns the number of terms in the series terminating with the soluble residual.

The derived series of the group G. The series is returned as a sequence of subgroups.

DerivedGroup(G) : GrpPerm -> GrpPerm

The derived subgroup of the group G.

An elementary abelian series is a chain of normal subgroups 1 < G_1 < ... < G_r = R <= G with the property that the quotient of each pair of successive terms in the series is elementary abelian and that there is no group R < H < G such that H/R is elementary abelian and H normal in G. The top of the series R is called the solvable radical and is the maximal normal solvable subgroup of G.

The Fitting subgroup of the group G.

Given a group G that is a p-group, return the Frattini subgroup.

Given a p-group G, return the Jennings series for G. The series is returned as a sequence of subgroups.

The lower central series of G. The series is returned as a sequence of subgroups.

A maximal normal subgroup of G.

The minimal normal subgroups of G. If G is primitive, this function is quite efficient. However, if G is intransitive or imprimitive, the minimal normal subgroups are found by computing the conjugacy classes of G and looking at the normal subgroups generated by each class. In this situation, the function is restricted to groups of moderate order.

The nilpotency class of the group G. If the group is not nilpotent, the value -1 is returned.

NormalClosure(G, H) : GrpPerm, GrpPerm -> GrpPerm

The normal closure of the subgroup H of group G.

The normal subgroup lattice of G. The subgroups are found by first determining the minimal normals using the conjugacy classes of G and then extending these layer by layer until G is reached. The algorithm is thus only practical in the case of small to medium sized groups.

The normal subgroups of G.

Given a soluble group G, and a prime p dividing |G|, return the lower p-central series for G. The series is returned as a sequence of subgroups.

SolubleRadical(G) : GrpPerm -> GrpPerm

SolvableRadical(G) : GrpPerm -> GrpPerm

Given a group G, return the maximal normal solvable subgroup of G.

Given a group G, compute a representation of the quotient G/R where R is the (solvable) radical of G. The resulting representation has the same degree as G. Both the permutation group Q isomorphic to G/R and a homomorphism phi: G -> Q are returned.

SolvableResidual(G) : GrpPerm -> GrpPerm

The solvable residual of the group G.

Given a group G and a subnormal subgroup H of G, return a sequence of subgroups commencing with G and terminating with H, such that each subgroup is normal in the previous one. If H is not subnormal in G, the empty sequence is returned.

The upper central series of G. The series is returned as a sequence of subgroups commencing with the trivial subgroup. Since the algorithm used requires the conjugacy classes of G, this function is much more restricted in its range of application than DerivedSeries and LowerCentralSeries.

• We compute the various series in the wreath product of the symmetric group of degree 4 with the dihedral group of order 12 (a soluble group).

> G := WreathProduct(Sym(4), DihedralGroup(4));
> G;
Permutation group G acting on a set of cardinality 16
    (1, 5, 9, 13)(2, 6, 10, 14)(3, 7, 11, 15)(4, 8, 12, 16)
    (1, 13)(2, 14)(3, 15)(4, 16)(5, 9)(6, 10)(7, 11)(8, 12)
    (1, 2, 3, 4)
    (1, 2)
> DerivedSeries(G);
[
    Permutation group G acting on a set of cardinality 16
    Order = 2654208 = 2^15 * 3^4
        (1, 5, 9, 13)(2, 6, 10, 14)(3, 7, 11, 15)(4, 8, 12, 16)
        (1, 13)(2, 14)(3, 15)(4, 16)(5, 9)(6, 10)(7, 11)(8, 12)
        (1, 2, 3, 4)
        (1, 2),
    Permutation group acting on a set of cardinality 16
    Order = 331776 = 2^12 * 3^4
        (1, 9)(2, 10)(3, 11)(4, 12)(5, 13)(6, 14)(7, 15)(8, 16)
        (1, 12)(2, 9)(3, 10)(4, 11)(5, 13)(6, 14)(7, 15)(8, 16)
        (2, 4, 3)(9, 11, 12)
        (6, 7, 8)(14, 16, 15)
        (5, 6, 7, 8)(9, 12, 11, 10)
        (6, 8, 7)
        (9, 12, 11, 10)(13, 14, 15, 16)
        (10, 11, 12)
        (2, 3)(9, 10, 11, 12)
        (3, 4)(7, 8)
        (2, 3, 4)(6, 8, 7)
        (3, 4)(6, 7, 8)(15, 16)
        (3, 4)(11, 12),
    Permutation group acting on a set of cardinality 16
    Order = 41472 = 2^9 * 3^4
        (5, 7)(6, 8)(13, 15)(14, 16)
        (1, 3)(2, 4)(9, 11)(10, 12)
        (1, 4)(2, 3)(9, 11)(10, 12)
        (2, 3, 4)(10, 12, 11)
        (3, 4)(6, 7)(10, 12)(15, 16)
        (6, 8, 7)(14, 15, 16)
        (6, 7, 8)(10, 12, 11)
        (2, 4)(7, 8)(9, 12, 11, 10)(14, 15)
        (6, 7, 8)(13, 16, 15)
        (10, 11, 12),
    Permutation group acting on a set of cardinality 16
    Order = 20736 = 2^8 * 3^4
        (1, 2)(3, 4)(9, 12)(10, 11)
        (5, 8)(6, 7)(13, 14)(15, 16)
        (9, 10)(11, 12)
        (5, 6)(7, 8)
        (1, 4)(2, 3)(9, 10)(11, 12)
        (13, 16)(14, 15)
        (9, 11)(10, 12)
        (13, 14)(15, 16)
        (6, 7, 8)(14, 16, 15)
        (10, 12, 11)
        (2, 4, 3)(10, 12, 11)
        (6, 7, 8)(10, 12, 11),
    Permutation group acting on a set of cardinality 16
    Order = 256 = 2^8
        (13, 14)(15, 16)
        (5, 6)(7, 8)
        (9, 10)(11, 12)
        (5, 7)(6, 8)(13, 14)(15, 16)
        (13, 15)(14, 16)
        (1, 3)(2, 4)(9, 10)(11, 12)
        (1, 4)(2, 3)(9, 10)(11, 12)
        (9, 12)(10, 11),
    Permutation group acting on a set of cardinality 16
    Order = 1
        Id($)
]
> DerivedLength(G);
5
> [ FactoredOrder(H) : H in LowerCentralSeries(G) ];
[
    [ <2, 15>, <3, 4> ],
    [ <2, 12>, <3, 4> ],
    [ <2, 10>, <3, 4> ],
    [ <2, 9>, <3, 4> ],
    [ <2, 8>, <3, 4> ]
]
> NilpotencyClass(G);
-1
> Centre(G);
Permutation group acting on a set of cardinality 16
Order = 1
    Id($)
> pCentralSeries(G, 2);
[
    Permutation group G acting on a set of cardinality 16
    Order = 2654208 = 2^15 * 3^4
        (1, 5, 9, 13)(2, 6, 10, 14)(3, 7, 11, 15)(4, 8, 12, 16)
        (1, 13)(2, 14)(3, 15)(4, 16)(5, 9)(6, 10)(7, 11)(8, 12)
        (1, 2, 3, 4)
        (1, 2),
    Permutation group acting on a set of cardinality 16
    Order = 331776 = 2^12 * 3^4
        (1, 9)(2, 10)(3, 11)(4, 12)(5, 13)(6, 14)(7, 15)(8, 16)
        (1, 2, 3, 4)(5, 8, 7, 6)
        (1, 2)(5, 6)
        (1, 2, 3, 4)(13, 16, 15, 14),
    Permutation group acting on a set of cardinality 16
    Order = 82944 = 2^10 * 3^4
        (1, 2, 3, 4)(5, 7)(6, 8)(9, 10, 11, 12)
        (1, 3)(2, 4)(5, 8, 7, 6)(13, 16, 15, 14)
        (1, 2)(9, 10)
        (1, 2)(5, 6)(9, 10)(13, 14)
        (1, 2, 3)
        (5, 6, 7),
    Permutation group acting on a set of cardinality 16
    Order = 41472 = 2^9 * 3^4
        (1, 2, 3, 4)(5, 6, 7, 8)(9, 12, 11, 10)(13, 16, 15, 14)
        (1, 2, 3)(5, 7, 6)
        (1, 4, 3, 2)(5, 6, 7, 8)(9, 10, 11, 12)(13, 16, 15, 14)
        (5, 6, 7)(9, 11, 10)
        (1, 2, 3)(13, 15, 14)
        (1, 2)(5, 6)(9, 10)(13, 14),
    Permutation group acting on a set of cardinality 16
    Order = 20736 = 2^8 * 3^4
        (1, 3)(2, 4)(9, 11)(10, 12)
        (1, 2, 3)(5, 6, 7)(9, 10, 11)
        (5, 6, 7)(9, 10, 11)(13, 14, 15)
        (1, 3, 2)(5, 7, 6)(13, 15, 14)
        (1, 2, 4)
        (5, 7)(6, 8)(13, 15)(14, 16)
]

The elementary abelian regular normal subgroup (EARNS) of the primitive group G. If G does not have a proper EARNS, then the trivial subgroup is returned.

The socle of the group G.

A simple factor of the socle of the primitive group G.

The set of simple factors of the socle of the primitive group G.

A chain of subgroups S_1, S_1 x S_2, ..., S_1 x ... x S_r, where S_1, ..., S_r are the simple factors of the socle of the primitive group G.

• We now examine the normal structure of a primitive group, the primitive-wreath product of Sym(5) and Sym(3) (with product action).

> G := PrimitiveWreathProduct(Sym(5), Sym(3));
> FactoredOrder(G);
[ <2, 10>, <3, 4>, <5, 3> ]
> E := EARNS(G);
> E;
Permutation group E acting on a set of cardinality 125
Order = 1
> DerivedSeries(G);
[
    Permutation group G acting on a set of cardinality 125
    Order = 10368000 = 2^10 * 3^4 * 5^3

    Permutation group acting on a set of cardinality 125
    Order = 2592000 = 2^8 * 3^4 * 5^3,

    Permutation group acting on a set of cardinality 125
    Order = 864000 = 2^8 * 3^3 * 5^3,

    Permutation group S acting on a set of cardinality 125
    Order = 216000 = 2^6 * 3^3 * 5^3
]
> S := Socle(G);
> S;
Permutation group S acting on a set of cardinality 125
Order = 216000 = 2^6 * 3^3 * 5^3
> Q := SocleFactors(G);
> Q;
[
    Permutation group acting on a set of cardinality 125
    Order = 60 = 2^2 * 3 * 5,

    Permutation group acting on a set of cardinality 125
    Order = 60 = 2^2 * 3 * 5,

    Permutation group acting on a set of cardinality 125
    Order = 60 = 2^2 * 3 * 5
]
> R := SocleSeries(G);
> R;
[
    Permutation group acting on a set of cardinality 125
    Order = 60 = 2^2 * 3 * 5,

    Permutation group acting on a set of cardinality 125
    Order = 3600 = 2^4 * 3^2 * 5^2,

    Permutation group acting on a set of cardinality 125
    Order = 216000 = 2^6 * 3^3 * 5^3
]

centerline{hfill vbox to 15truecm{vfill offinterlineskip hrule halign{&vrule#& hfil#hfil &vrule#& #hfil cr &f&&hfil Family name&cr noalign{hrule} & 1 && A(d, q) &cr & 2 && B(d, q) &cr & 3 && C(d, q) &cr & 4 && D(d, q) &cr & 5 && G(2, q) &cr & 6 && F(4, q) &cr & 7 && E(6, q) &cr & 8 && E(7, q) &cr & 9 && E(8, q) &cr & 10 && 2A(d, q) &cr & 11 && 2B(2, q) &cr & 12 && 2D(d, q) &cr & 13 && 3D(4, q) &cr & 14 && 2G(2, q) &cr & 15 && 2F(4, q) &cr & 16 && 2E(6, q) &cr & 17 && Alternating(d) &cr & 18 && Sporadic group --- see Table 2. & cr & 19 && Cyclic(q) &cr }hrule vfill}hfill vbox to 15truecm{vfill offinterlineskip hrule halign{&vrule#& hfil#hfil &vrule#& #hfil cr &d&&hfil Group name&cr noalign{hrule} & 1 && M_(11) &cr & 2 && M_(12) &cr & 3 && M_(22) &cr & 4 && M_(23) &cr & 5 && M_(24) &cr & 6 && J_1 &cr & 7 && HS &cr & 8 && J_2 &cr & 9 && MCL &cr & 10 && SUZ &cr & 11 && J_3 &cr & 12 && CO_1 &cr & 13 && CO_2 &cr & 14 && CO_3 &cr & 15 && HE &cr & 16 && M(22) &cr & 17 && M(23) &cr & 18 && M(24) &cr & 19 && LY &cr & 20 && RU &cr & 21 && ON &cr & 22 && TH &cr & 23 && HA &cr & 24 && BM &cr & 25 && M &cr & 26 && J_4 &cr }hrule vfill}hfill} centerline hfill{Table 1: Family numbers and names hfill{Table 2: Sporadic groups}hfill}

AbelianBasis(G) : GrpPerm -> [ GrpPermElt ], [ RngIntElt ]

Given an abelian group G, return sequence B and I such that ( order)(B[i]) = I[i] and < B > = G and I are the types of each p-primary component of G.

Given an abelian group G, return a sequence Q containing the types of each p-primary component of G.

Given a group G, return a sequence of the isomorphism types <f, d, q, m> of the chief factors. An isomorphism type in a chief factor should be understood as the direct product of m copies of the simple group described by <f, d, q> (see CompositionFactors below).

Given a group G, return the chief series of G and a sequence of the corresponding isomorphism types <f, d, q, m> of the chief factors. An isomorphism type in a chief factor should be understood as the direct product of m copies of the simple group described by <f, d, q> (see CompositionFactors below).

Given a permutation group G, return a sequence S of tuples that represent the composition factors of G, ordered according to some composition series of G. Each tuple is a triple of integers f, d, q that defines the isomorphism type of the corresponding composition factor. A triple < f, d, q > describes a simple group as follows. The integer f defines the family to which the group belongs, and d and q are the parameters of the family. The length of the sequence S is the number of composition factors of G.

The families are:
    f      family name
-------------------------
    1       A(d, q) 
    2       B(d, q) 
    3       C(d, q) 
    4       D(d, q) 
    5       G(2, q) 
    6       F(4, q) 
    7       E(6, q) 
    8       E(7, q) 
    9       E(8, q) 
   10       2A(d, q) 
   11       2B(2, q) 
   12       2D(d, q) 
   13       3D(4, q) 
   14       2G(2, q) 
   15       2F(4, q) 
   16       2E(6, q) 
   17       Alternating(d) 
   18       Sporadic group --- see next list
   19       Cyclic(q) 

For f=18, the sporadic groups are:
    d      group name
-------------------------    
    1      M_11
    2      M_12
    3      M_22
    4      M_23 
    5      M_24
    6      J_1
    7      HS
    8      J_2
    9      MCL
   10      SUZ
   11      J_3
   12      CO_1
   13      CO_2
   14      CO_3
   15      HE
   16      M(22)
   17      M(23)
   18      M(24)
   19      LY
   20      RU
   21      ON
   22      TH
   23      HA
   24      BM
   25      M
   26      J_4

Given a simple group G, determine the isomorphism type of G. The type is returned in the form of a triple of three integers f, d and q, where the interpretation of these integers is that given in the description of the function CompositionFactors.

• We illustrate the function CompositionFactors by applying it to the group associated with Rubik's cube.

> G := sub<Sym(48) |
>     (1,3,8,6)(2,5,7,4)(9,48,15,12)(10,47,16,13)(11,46,17,14),
>     (6,15,35,26)(7,22,34,19)(8,30,33,11)(12,14,29,27)(13,21,28,20),
>     (1,12,33,41)(4,20,36,44)(6,27,38,46)(9,11,26,24)(10,19,25,18),
>     (1,24,40,17)(2,18,39,23)(3,9,38,32)(41,43,48,46)(42,45,47,44),
>     (3,43,35,14)(5,45,37,21)(8,48,40,29)(15,17,32,30)(16,23,31,22),
>     (24,27,30,43)(25,28,31,42)(26,29,32,41)(33,35,40,38)(34,37,39,36)
>         >;
> FactoredOrder(G);
[ <2, 27>, <3, 14>, <5, 3>, <7, 2>, <11, 1> ]
> CompositionFactors(G);
     G
     |  Cyclic(2)
     *
     |  Alternating(12)
     *
     |  Cyclic(2)
     *
     |  Cyclic(2)
     *
     |  Cyclic(2)
     *
     |  Cyclic(2)
     *
     |  Cyclic(2)
     *
     |  Cyclic(2)
     *
     |  Cyclic(2)
     *
     |  Cyclic(2)
     *
     |  Cyclic(2)
     *
     |  Cyclic(2)
     *
     |  Cyclic(2)
     *
     |  Alternating(8)
     *
     |  Cyclic(3)
     *
     |  Cyclic(3)
     *
     |  Cyclic(3)
     *
     |  Cyclic(3)
     *
     |  Cyclic(3)
     *
     |  Cyclic(3)
     *
     |  Cyclic(3)
     1