Building Permutation Groups

Examples of permutation groups are routinely constructed by taking one or more standard groups and applying some extension procedure to construct a group having the given groups as subgroups or quotient groups. In the first subsection we describe functions which construct some well-known groups and in the following subsection we give functions for constructing direct and wreath products.

Some Standard Permutation Groups

A number of functions are provided which construct various standard groups. The effect of these functions is to construct the group on some standard set of generating permutations.

AbelianGroup(GrpPerm, Q): Cat, [ RngIntElt ] -> GrpPerm: Construct the abelian group defined by the sequence \(Q = [n_1,\ldots,n_r]\) of positive integers. The function constructs the direct product of cyclic groups

\[Z(n_1) \times Z(n_2) \times \cdots \times Z(n_r).\]

AlternatingGroup(GrpPerm, n): Cat, RngIntElt -> GrpPerm
AlternatingGroup(n): RngIntElt -> GrpPerm
Alt(n): RngIntElt -> GrpPerm: Construct the alternating group of degree \(n\) on generators \((3,4,\ldots,n)\) and \((1,2,3)\), if \(n\) is odd, or \((1,2)(3,4\), \(\ldots\), \(n)\) and \((1,2,3)\), if \(n\) is even.

CyclicGroup(GrpPerm, n): Cat, RngIntElt -> GrpPerm
CyclicGroup(n): RngIntElt -> GrpPerm: Construct the cyclic group of order \(n\) with generator \((1,2,\ldots,n)\).

DihedralGroup(GrpPerm, n): Cat, RngIntElt -> GrpPerm
DihedralGroup(n): RngIntElt -> GrpPerm: Construct the dihedral group of degree \(n\) and order \(2*n\) on generators \((1,2,\ldots,n)\) and \((1,n)(2,n-1)\cdots\).

Sym(GrpPerm, n): Cat, RngIntElt -> GrpPerm
SymmetricGroup(GrpPerm, n): Cat, RngIntElt -> GrpPerm
Sym(n): RngIntElt -> GrpPerm
SymmetricGroup(n): RngIntElt -> GrpPerm: Construct the symmetric group of degree \(n\) on generators \((1,2,\ldots,n)\) and \((1,2)\).

ExtraSpecialGroup(GrpPerm, p, n : parameters): Cat, RngIntElt, RngIntElt -> GrpPerm

ExtraSpecialGroup(p, n : parameters): RngIntElt, RngIntElt -> GrpPerm

Type: MonStgElt                    Default: "+"

Given a small prime \(p\) and a small positive integer \(n\), construct an extra-special group \(G\) of order \(p^{2n+1}\) in the category GrpPerm. The isomorphism type of \(G\) can be selected using the parameter Type.

Possible values for this parameter are “\(+\)” (default) and “\(-\)“.

If Type is set to “\(+\)”, the function returns for \(p = 2\) the central product of \(n\) copies of the dihedral group of order 8, and for \(p > 2\) it returns the unique extra-special group of order \(p^{2n+1}\) and exponent \(p\).

If Type is set to “\(-\)”, the function returns for \(p = 2\) the central product of a quaternion group of order 8 and \(n-1\) copies of the dihedral group of order 8, and for \(p > 2\) it returns the unique extra-special group of order \(p^{2n+1}\) and exponent \(p^2\).

YoungSubgroup(L): [RngIntElt] -> GrpPerm

Full: RngIntElt                    Default: false

Given a sequence \(L\) of positive integers, compute the Young subgroup parameterized by \(L\), i.e., the direct product of the symmetric groups on \(L_i\) points. If the optional parameter Full is given, construct the group as a subgroup of the symmetric group on Full elements.

Example: Standard Permutation Groups (ex-98baac)

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The abelian group \(Z_2 \times Z_2 \times Z_4\):

> A := AbelianGroup(GrpPerm, [2, 2, 4] );
> A;
Permutation group A acting on a set of cardinality 8
Order = 16 = 2^4
    (1, 2)
    (3, 4)
    (5, 6, 7, 8)

The alternating group of degree \(12\):

> A12 := AlternatingGroup(GrpPerm, 12);
> A12;
Permutation group A12 acting on a set of cardinality 12
Order = 239500800 = 2^9 * 3^5 * 5^2 * 7 * 11
    (1, 2)(3, 4, 5, 6, 7, 8, 9, 10, 11, 12)
    (1, 2, 3)

The cyclic group \(Z_{24}\):

> Z24 := CyclicGroup(GrpPerm, 24);
> Z24;
Permutation group Z24 on a set of cardinality 24
Order = 24 = 2^3 * 3
    (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
        16, 17, 18, 19, 20, 21, 22, 23, 24)

The dihedral group of order \(24\):

> D12 := DihedralGroup(GrpPerm, 12);
> D12;
Permutation group D12 acting on a set of cardinality 12
Order = 24 = 2^3 * 3
    (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)
    (1, 12)(2, 11)(3, 10)(4, 9)(5, 8)(6, 7)

The symmetric group of degree \(8\):

> S8 := SymmetricGroup(GrpPerm, 8);
> S8;
Symmetric group S8 acting on a set of cardinality 8
Order = 40320 = 2^7 * 3^2 * 5 * 7

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Direct Products and Wreath Products

DirectProduct(G, H): GrpPerm, GrpPerm -> GrpPerm, [ Hom(Grp) ], [ Hom(Grp) ]: Given two permutation groups \(G\) and \(H\), construct the direct product \(D\) of \(G\) and \(H\) as an intransitive group having degree equal to the sum of the degrees of \(G\) and \(H\). In addition, the sequences \(I\) of inclusions and \(P\) of projections are returned, satisfying \(I[i]: K_i\rightarrow D(K_i)\) and \(P[i]: D\rightarrow K_i\) (where \(K_1 = G, K_2 = H\) and \(D(K)\) is the group \(K\) represented naturally as a subgroup of \(D\)).

DirectProduct(Q): [ GrpPerm ] -> GrpPerm, [ Hom(Grp) ], [ Hom(Grp) ]: Given a sequence \(Q\) of \(n\) permutation groups, construct the direct product \(Q[1] \times Q[2] \times \ldots \times Q[n]\) as an intransitive group of degree equal to the sum of the degrees of the groups \(Q[i]\), \((i = 1, \ldots, n)\). In addition, the sequences \(I\) of inclusion and \(P\) of projections are returned, satisfying \(I[i]: Q[i]\rightarrow D(Q[i])\) and \(P[i]: D\rightarrow Q[i]\) (where \(D(K)\) is the group \(K\) represented naturally as a subgroup of \(D\)).

PrimitiveWreathProduct(G, H): GrpPerm, GrpPerm -> GrpPerm: Given permutation groups \(G\) and \(H\), construct the wreath product \(G\wr H\) of \(G\) and \(H\), where \(G\wr H\) has product action.

PrimitiveWreathProduct(Q): [ GrpPerm ] -> GrpPerm: Given a sequence \(Q\) of \(n\) permutation groups, construct the iterated wreath product \(T = ( \ldots (Q[1] \wr Q[2]) \wr \ldots \wr Q[n])\), where \(T\) has product action.

WreathProduct(G, H): GrpPerm, GrpPerm -> GrpPerm, SeqEnum[Map], Map, Map: Given permutation groups \(G\) and \(H\), construct the wreath product \(W = G\wr H\) of \(G\) and \(H\), where \(G\wr H\) has imprimitive action. The function also returns the sequence of Degree(H) inclusions of \(G\) into \(W\), the inclusion of \(H\) into \(W\), and the projection of \(W\) onto \(H\).

WreathProduct(Q): [ GrpPerm ] -> GrpPerm: Given a sequence \(Q\) of \(n\) permutation groups, construct the iterated wreath product \(W = ( \ldots (Q[1] \wr Q[2]) \wr \ldots \wr Q[n])\), where \(W\) has imprimitive action.

WreathProduct(B): GSet -> GrpPerm, GrpPerm, GrpPerm: Given a block system \(B\) of some permutation group \(G\), compute the wreath-product corresponding to \(B\).

WreathProduct(G, B): GrpPerm, GSet -> GrpPerm, GrpPerm, GrpPerm: Compute the smallest wreath product \(W\) to the block system \(B\) of \(G\) such that \(G \subseteq W\). Also return the complement as a subgroup of \(W\). The third parameter is a subgroup which is isomorphic to the action within a block.

Example: Direct and Wreath Products (ex-4e2871)

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We define \(G\) to be the symmetric group of degree \(4\) and \(H\) to be the dihedral group of order \(8\). We then proceed to form the direct, primitive-wreath, and wreath products of \(G\) and \(H\).

> G := SymmetricGroup(GrpPerm, 4);
> H := DihedralGroup(GrpPerm, 3);
> D := DirectProduct(G, H);
> D;
Permutation group D acting on a set of cardinality 7
Order = 144 = 2^4 * 3^2
    (1, 2, 3, 4)
    (1, 2)
    (5, 6, 7)
    (5, 6)
> T := PrimitiveWreathProduct(G, H);
> T;
Permutation group T acting on a set of cardinality 64
Order = 82944 = 2^10 * 3^4
    (2, 5, 17)(3, 9, 33)(4, 13, 49)(6, 21, 18)(7, 25, 34)(8, 29, 50)
       (10, 37, 19) (11, 41, 35)(12, 45, 51) (14, 53, 20)(15, 57, 36)
       (16, 61, 52)(23, 26, 38) (24, 30, 54)(27, 42, 39)(28, 46, 55)
       (31, 58, 40) (32, 62, 56)(44, 47, 59)(48, 63, 60)
    (2, 5)(3, 9)(4, 13)(7, 10)(8, 14)(12, 15)(18, 21)(19 , 25)(20, 29)
       (23, 26)(24, 30)(28, 31)(34, 37)(35 , 41)(36, 45)(39, 42)(40, 46)
       (44, 47)(50, 53)(51 , 57)(52, 61)(55, 58)(56, 62)(60, 63)
    (1, 2, 3, 4)(5, 6, 7, 8)(9, 10, 11, 12)(13, 14, 15, 16)(17, 18, 19, 20)
       (21, 22, 23, 24)(25, 26, 27, 28)(29, 30, 31, 32)(33, 34, 35, 36)
       (37, 38, 39, 40)(41, 42, 43, 44)(45, 46, 47, 48)(49, 50, 51, 52)
       (53, 54, 55, 56)(57, 58, 59, 60)(61, 62, 63, 64)
    (1, 2)(5, 6)(9, 10)(13, 14)(17, 18)(21, 22)(25, 26)( 29, 30)(33, 34)
       (37, 38)(41, 42)(45, 46)(49, 50)( 53, 54)(57, 58)(61, 62)
> W := WreathProduct(G, H);
> W;
Permutation group W acting on a set of cardinality 12
Order = 82944 = 2^10 * 3^4
    (1, 5, 9)(2, 6, 10)(3, 7, 11)(4, 8, 12)
    (1, 5)(2, 6)(3, 7)(4, 8)
    (1, 2, 3, 4)
    (1, 2)

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Building Permutation Groups#

Some Standard Permutation Groups#

Direct Products and Wreath Products#

Building Permutation Groups

Some Standard Permutation Groups

Direct Products and Wreath Products