Let G be a group. A G-set is a pair (Y, f), where Y is a set and f : Y x G -> Y is a mapping such that (a) f(f(y, g), h) = f(y, gh), for all g, h in G and (b) f(y, 1) = y, for all y in Y. The mapping f defines the action of G on the set Y.
If G is defined as a permutation group acting on the set X and Y is another G-set then there is a homomorphism of G^(X) into G^(Y).
We distinguish three types of G-set for a permutation group G. The set on which G is defined will be referred to as the natural G-set and the action of G on X as the natural action of G.
Let A be some set. A derived set of A is defined (recursively) as follows:
Finally, a general G-set is an arbitrary set Y with an action f satisfying the conditions (a) and (b).
The notion of a G-set enables the user to work with several different
actions of G. Rather than having to always work with the image
of G with respect to an action on a set Y, the user may specify
the required operation in terms of G.
Creating a G-Set
GSet(G, Y) : GrpPerm, SetIndx -> GSet
Given a group G and an indexed set Y with the same cardinality as the natural G-set, return a G-set corresponding to the natural bijection between the labelling L of G and Y. Explicitly, the bijection is f: L -> Y: l |-> Y[( Position)(L, l)]. The the returned G-set is the set Y endowed with the action phi: Y x G -> Y: (y, p) |-> f(p(f^(-1)(y))).
Return the smallest derived G-set containing Y under the action given by X. If X is omitted, then the natural action will be assumed, and Y cannot be an indexed set. In practice, the set Y is expanded until for each element y of the expanded Y, the image of y under each generator of G and the action describe by X is also in Y. The action of Y is then just the derived action of X.
Given a permutation group G, return the G-set corresponding to the natural action of G.
Construct the smallest G-set containing Y with the given action f. The map f must satisfy the requirements of a G-set action. In particular, the domain of f must be a superset of Y x G, the codomain a superset of Y and the two conditions listed at the beginning of this section must be met.
The map giving the action of the group on the G-set Y.
The group associated with the G-set Y.
Given a permutation group G of degree n, return an indexed set giving the internal mapping of the natural G-set of G onto the set { 1, ..., n }, where n is the degree of G.
Given an element g of a permutation group G and a G-set Y, return the cardinality of the subset of Y consisting of points that are moved by g. If Y is omitted, the natural G-set X is assumed.
Given a G-set Y, return the cardinality of Y. If Y is omitted, the natural G-set X is assumed.
Given an element g of a permutation group G and a G-set Y, return the subset of Y consisting of points that are moved by g. If Y is omitted, the natural G-set X is assumed.
Given a permutation group G and a G-set Y, return the subset of Y consisting of points that are moved by at least one element of G. If Y is omitted the natural G-set for G is assumed.
Given a permutation group G with natural G-set X and an object x which is an element of some derived G-set of X, find the image of x under G.
Given a permutation group G, a G-set Y, and an element y of Y, find the image of y under G. If y is an element of some derived G-set of G, the set Y may be omitted.
Given a permutation g belonging to a group G and a G-set Y, construct the fixed-point set of g in its action on Y. In the case in which Y is the natural G-set, Y may be omitted. The fixed-point set is returned as a subset of points of Y.
The fixed-point set of the permutation group G in its action on the G-set Y (or the natural G-set for G if Y is omitted).
Given a permutation group G with natural G-set X and an element x belonging to some derived G-set of X, construct the orbit of x under G. The orbit is returned as a G-set.
Given a permutation group G, a G-set Y, and an element y belonging to Y, construct the orbit of y under G. The orbit is returned as a G-set. If y is an element of some derived G-set of G, the set Y may be omitted.
Given a permutation group G and a G-set Y, construct the orbits of G on Y. If the set Y is omitted, the orbits of G on its natural G-set are constructed. The orbits are returned as a sequence of G-sets.
Given a subset S of the G-set Y, construct the smallest G-invariant subset of Y that contains S. If Y is the natural G-set for G it may be omitted.
Given elements y and z belonging either to a G-set Y or to a (restricted) derived set of Y, return the value true if there exists an element g in G such that y^g = z. Otherwise, return false. If such an element exists, then it is returned as the second value of the function. If y and z belong to the natural G-set, then Y may be omitted. Currently, y and z are restricted to being elements, sets of elements, sequences of elements, or ordered partitions of Y.
Given a permutation group G and a G-set Y, and an object y which is either an element, a sequence of elements, a set of elements, a partition or a tuple over the G-set Y, find the stabilizer of y in G. The stabilizer is returned as a subgroup of G. If Y is the natural G-set, it may be omitted.
True if G acts primitively on the G-set Y. If Y is the natural G-set, the set Y may be omitted.
True if G acts transitively on the G-set Y. If Y is the natural G-set, the set Y may be omitted.
True if G acts k-transitively on the G-set Y. If Y is the natural G-set, the set Y may be omitted.
True if G acts sharply k-transitively on the G-set Y. If Y is the natural G-set, the set Y may be omitted.
The degree of transitivity of G acting on the G-set Y. The set Y may be omitted if it is the same as the natural G-set.
True if G acts regularly on the G-set Y. If Y is the natural G-set, the set Y may be omitted.
True if G acts semiregularly on the G-set Y. If Y is the natural G-set, the set Y may be omitted.
Given a permutation group G, a G-set Y for G, and a union of orbits S for G in its action on Y, return true if G acts semiregularly on S. If Y is the natural G-set, then Y may be omitted.
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,24), (2,16,9,6,8)(3,12,13,18,4)(7,17,10,11,22)(14,19,21,20,15), (1,22)(2,11)(3,15)(4,17)(5,9)(6,19)(7,13)(8,20)(10,16)(12,21)(14,18)(23,24).
> M24 := sub< Sym(24) |
> (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,24),
> (2,16,9,6,8)(3,12,13,18,4)(7,17,10,11,22)(14,19,21,20,15),
> (1,22)(2,11)(3,15)(4,17)(5,9)(6,19)(7,13)(8,20)(10,16)(12,21)(14,18)(23,24)>;
> M24;
Permutation group M24 acting on a set of cardinality 24
(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 24)
(2, 16, 9, 6, 8)(3, 12, 13, 18, 4)(7, 17, 10, 11, 22)(14, 19, 21, 20, 15)
(1, 22)(2, 11)(3, 15)(4, 17)(5, 9)(6, 19)(7, 13)(8, 20)(10, 16)(12, 21)
(14, 18)(23, 24)
> /*
> We take a random element x and use it to compute some images
> */
> x := Random(M24);
> 1^x;
7
> [1,2,3,4]^x;
[ 7, 9, 8, 17 ]
> { 1,2,3,4 }^x;
{ 17, 7, 8, 9 }
> /*
> We compute the stabilizer of the point 1, which is the group M23
> */
> S1 := Stabilizer(M24, 1);
> S1;
Permutation group S1 acting on a set of cardinality 24
Order = 10200960 = 2^7 * 3^2 * 5 * 7 * 11 * 23
(2, 16, 9, 6, 8)(3, 12, 13, 18, 4)(7, 17, 10, 11, 22)(14, 19, 21, 20, 15)
(7, 17, 22)(8, 11, 13)(9, 14, 12)(10, 20, 19)(15, 23, 18)(16, 21, 24)
(3, 6, 18)(5, 16, 14)(7, 21, 22)(8, 19, 17)(9, 20, 24)(11, 12, 13)
(6, 18, 15)(7, 19, 16)(8, 13, 11)(9, 10, 22)(12, 21, 20)(14, 17, 24)
(4, 12, 6, 19)(5, 22, 24, 8)(7, 17, 20, 14)(9, 15, 13, 18)(10, 21)(11, 16)
(6, 22, 7)(8, 13, 11)(9, 20, 16)(10, 18, 21)(12, 15, 19)(14, 24, 23)
(5, 12, 21)(6, 15, 18)(7, 22, 8)(9, 16, 17)(10, 14, 13)(11, 24, 19)
> /*
> We compute the stabilizer of the sequence [1,2,3,4,5]
> */
> SQ := Stabilizer(M24, [1,2,3,4,5]);
> SQ;
Permutation group SQ acting on a set of cardinality 24
Order = 48 = 2^4 * 3
(6, 18, 15)(7, 19, 16)(8, 13, 11)(9, 10, 22)(12, 21, 20)(14, 17, 24)
(7, 17, 22)(8, 11, 13)(9, 14, 12)(10, 20, 19)(15, 23, 18)(16, 21, 24)
(6, 22, 7)(8, 13, 11)(9, 20, 16)(10, 18, 21)(12, 15, 19)(14, 24, 23)
> Orbits(SQ);
[
GSet{ 1 },
GSet{ 2 },
GSet{ 3 },
GSet{ 4 },
GSet{ 5 },
GSet{ 6,18, 22, 15, 21, 9, 7, 23, 19, 20, 24, 10,
14, 17, 16 12 },
GSet{ 8, 13, 11 }
]
> /*
> The five fixed points together with the orbit of length 3 form a block of
> a 5-(24,8,1) design. By computing the orbit of this block under M24, we
> obtain all the blocks of the design.
> */
> B := { 1,2,3,4,5,8,11,13 };
> D := B^M24;
> #D;
759
> /*
> Finally, we compute the stabilizer of the set { 1,2,3,4,5 }
> */
> SS := Stabilizer(M24, { 1,2,3,4,5 });
> SS;
Permutation group SS acting on a set of cardinality 24
Order = 5760 = 2^7 * 3^2 * 5
(1, 2)(7, 22)(9, 16)(10, 19)(11, 13)(12, 21)(14, 24)(15, 18)
(2, 3)(7, 24)(9, 14)(11, 13)(15, 23)(16, 22)(17, 21)(19, 20)
(3, 4)(7, 19)(10, 22)(11, 13)(12, 14)(15, 18)(17, 20)(21, 24)
(4, 5)(7, 22)(9, 14)(10, 18)(11, 13)(15, 19)(16, 24)(20, 23)
(7, 22, 17)(8, 13, 11)(9, 12, 14)(10, 19, 20)(15, 18, 23)(16, 24, 21)
(6, 12)(7, 23)(9, 14)(10, 18)(15, 24)(16, 19)(17, 21)(20, 22)
(6, 18)(7, 24)(9, 17)(10, 12)(14, 21)(15, 23)(16, 20)(19, 22)
(6, 14)(7, 16)(9, 12)(10, 17)(15, 22)(18, 21)(19, 23)(20, 24)
(6, 15)(7, 10)(9, 20)(12, 24)(14, 22)(16, 17)(18, 23)(19, 21)
Given a permutation group G defined to be acting on X and a set Y, construct the homomorphism phi: G -> L, where the permutation group L gives the action of G on the set Y. The function returns:
- The natural homomorphism phi: G -> L;
- The induced group L;
- The kernel of the action (a subgroup of G).
Given a permutation group G defined to be acting on X and a set Y, construct the permutation group L giving the action of G on the set Y.
Construct the kernel of the homomorphism phi : G -> L, where the permutation group L gives the action of G on the G-set Y.
True if the action of G on the G-set Y is faithful.
> G := ProjectiveSpecialLinearGroup(3, 4);
> O2 := pCore( Stabilizer(G, 1), 2 );
> O2;
Permutation group O2 acting on a set of cardinality 21
Order = 16 = 2^4
(3, 4)(5, 7)(9, 16)(10, 17)(11, 15)(13, 18)(14, 19)(20, 21)
(3, 20)(4, 21)(5, 15)(7, 11)(9, 10)(13, 19)(14, 18)(16, 17)
(2, 8)(5, 15)(6, 12)(7, 11)(9, 17)(10, 16)(13, 18)(14, 19)
(2, 12)(5, 11)(6, 8)(7, 15)(9, 16)(10, 17)(13, 19)(14, 18)
> flag := < 1, Orbit(O2, 2) >;
> flag;
<1, GSet{ 2, 6, 8, 12 }>
> flags := GSet(G, Orbit(G, flag));
> #flags;
105
> GOnFlags := ActionImage(G, flags);
> GOnFlags;
Permutation group GOnFlags acting on a set of
cardinality 105
Order = 20160 = 2^6 * 3^2 * 5 * 7
> Stabilizer(GOnFlags, Rep(flags));
Permutation group acting on a set of cardinality 105
Order = 192 = 2^6 * 3
The operations described here are concerned with the class of G-sets
consisting of G-invariant subsets of the natural G-set. Because of
the special nature of such G-sets, more efficient algorithms are
available for computing with homomorphisms of G induced by the action
of G on such a G-set.
OrbitAction(G, T) : GrpPerm, GSet -> Hom(Grp), GrpPerm, GrpPerm
The homomorphism f : G -> L induced by the action of G on the G-invariant subset T of X (a union of orbits).
The group L defined by the action of G on the G-invariant subset T of X (a union of orbits).
The kernel of the homomorphism f : G -> L, where the group L gives the action of G on the G-invariant subset T of X (a union of orbits).
True if the subset S of Support(G) is invariant under G.
(3, 17, 26)(4, 16, 25)(5, 18, 27)(8, 15, 24), (1, 32, 10)(2, 31, 11)(3, 35, 12)(6, 30, 15), (12, 33, 24)(13, 29, 20)(14, 28, 19)(17, 30, 21), (6, 26, 33)(7, 22, 34)(8, 21, 35)(9, 23, 36).
> G := PermutationGroup< 36 | (3, 17, 26)(4, 16, 25)(5, 18, 27)(8, 15, 24),
> (1, 32, 10)(2, 31, 11)(3, 35, 12)(6, 30, 15),
> (12, 33, 24)(13, 29, 20)(14, 28, 19)(17, 30, 21),
> (6, 26, 33)(7, 22, 34)(8, 21, 35)(9, 23, 36) >;
> IsTransitive(G);
false
> Orbit(G, 1);
GSet{ 1, 32, 10 }
> O := Orbits(G);
> O;
[
GSet{ 1, 32, 10 },
GSet{ 2, 31, 11 },
GSet{ 3, 17, 35, 26, 30, 12, 8, 33, 15, 21, 24, 6 },
GSet{ 4, 16, 25 },
GSet{ 5, 18, 27 },
GSet{ 7, 22, 34 },
GSet{ 9, 23, 36 },
GSet{ 13, 29, 20 },
GSet{ 14, 28, 19 }
]
> Order(G);
933120
> /* Thus the group is intransitive having eight orbits of size 3 and one
> orbit of size 12. We consider the action of G on the orbit of size 12. */
> f := OrbitAction(G, O[3]);
> f;
Mapping from: GrpPerm: G to GrpPerm: $
> Im := Image(f);
> Im;
Permutation group acting on a set of cardinality 12
Order = 11520 = 2^8 * 3^2 * 5
(1, 6, 9)(3, 5, 8)
(4, 11, 8)(6, 10, 7)
(6, 8)(7, 11)
(2, 8, 10)(4, 12, 6)
(3, 10, 8)(4, 6, 9)
> Ker := Kernel(f);
> Ker;
Permutation group acting on a set of cardinality 36
Order = 81 = 3^4
(4, 16, 25)(5, 18, 27)
(7, 22, 34)(9, 23, 36)
(13, 29, 20)(14, 28, 19)
(1, 32, 10)(2, 31, 11)(4, 25, 16)(5, 27, 18)
> IsElementaryAbelian(Ker);
true
> /* Thus G has an elementary abelian normal subgroup of order 81
> which is the kernel of the restriction of G to the orbit of size 12. */
Given a transitive permutation group G with natural G-set X, and a subset S of X, return true if S is a block for G in its action on X.
True if the transitive permutation group G is primitive.
Construct a G-invariant partition P for the transitive permutation group G with natural G-set X. The partition P is maximal in the sense that there is no G-invariant partition P' of X such that some block of P' properly contains a block of the partition P. The block system is returned as a partition of X. If G is primitive, the empty set is returned.
Construct a non-trivial G-invariant partition P of the natural G-set X of the transitive permutation group G. The partition P is minimal in the sense that there is no G-invariant partition P' of X such that some block of P' is properly contained in some block of the partition P. The block system is returned as a partition of X. If G is primitive, or if no partition satisfying the side-conditions (see below) is found, then the empty set is returned.
Block = S: { Elt } Default: []If S is non-empty, then the partition P must possess a cell B such that S is a subset of B.
Construct all non-trivial minimal G-invariant partitions of the natural G-set X of the transitive permutation group G. A partition P is minimal in the sense that there is no G-invariant partition P' of X such that some block of P' is properly contained in some block of the partition P.
The minimal block systems are returned as a sequence of sets of sets. If G is primitive, the function returns the empty sequence.
Limit = n: RngIntElt Default: InfinityThe function will return after creating at most n block systems. This option is useful in situations where, say, two distinct minimal blocks systems are required for a reduction algorithm.
Given a transitive permutation group G with natural G-set X and a G-invariant partition P of X, construct the group L induced by the action of G on the blocks of P. The function returnsThe relationship between the supports of G and L is given by the labelling mapping.
- The natural homomorphism f: G -> L;
- The induced group;
- The kernel of the action (a subgroup of G).
Given a transitive permutation group G with natural G-set X and a G-invariant partition P of X, construct the group induced by the action of G on the blocks of P.
Given a transitive permutation group G with natural G-set X and a G-invariant partition P of X, construct the kernel of the action of G on the blocks of P.
> G := sub< Sym(100) |
> (1,21,41,61,81)(2,82,62,42,22)(3,23,43,63,83)(4,84,64,44,24)
> (5,25,45,65,85)(6,86,66,46,26)(7,27,47,67,87)(8,88,68,48,28)
> (9,29,49,69,89)(10,90,70,50,30)(11,31,51,71,91)(12,92,72,52,32)
> (13,33,53,73,93)(14,94,74,54,34)(15,35,55,75,95)(16,96,76,56,36)
> (17,37,57,77,97)(18,98,78,58,38)(19,39,59,79,99)(20,100,80,60,40),
> (1,4,6,7,10)(2,3,5,8,9)(11,19,17,15,14)(12,20,18,16,13)(21,24,26,27,30)
> (22,23,25,28,29)(31,39,37,35,34)(32,40,38,36,33)(41,44,46,47,50)
> (42,43,45,48,49)(51,59,57,55,54)(52,60,58,56,53)(61,64,66,67,70)
> (62,63,65,68,69)(71,79,77,75,74)(72,80,78,76,73)(81,84,86,87,90)
> (82,83,85,88,89)(91,99,97,95,94)(92,100,98,96,93),
> (1,11,2,12)(3,13,4,14)(5,16,6,15)(7,17,8,18)(9,20,10,19)(21,31,22,32)
> (23,33,24,34)(25,36,26,35)(27,37,28,38)(29,40,30,39)(41,51,42,52)
> (43,53,44,54)(45,56,46,55)(47,57,48,58)(49,60,50,59)(61,71,62,72)
> (63,73,64,74)(65,76,66,75)(67,77,68,78)(69,80,70,79)(81,91,82,92)
> (83,93,84,94)(85,96,86,95)(87,97,88,98)(89,100,90,99) >;
> MaxPart := MaximalPartition(G);
> #MaxPart;
2
> MaxPart;
GSet{
{ 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 71,
72, 73, 74, 75, 76, 77, 78, 79, 80, 91, 92, 93, 94, 95, 96, 97,
98, 99, 100 },
{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 61, 62, 63, 64,
65, 66, 67, 68, 69, 70, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90 }
}
> MinPart := MinimalPartition(G);
> #MinPart;
50
> /*
> Thus the group has a partition consisting of 50 blocks of size 2.
> */
> Parts := MinimalPartitions(G);
> [ #p : p in Parts ];
[ 50, 50, 50, 50, 20, 50 ]
> /*
> Thus the group has six distinct minimal G-invariant partitions,
> Of these five have 50 blocks of size two while the remaining one
> has 20 blocks of size 5. We examine the action of G on one of the
> partitions into 50 blocks of size 2.
> */
> f, Im, Ker := BlocksAction(G, Parts[1]);
> f;
Mapping from: GrpPerm: G to GrpPerm: Im
> Im;
Permutation group Im acting on a set of cardinality 50
Order = 7812500 = 2^2 * 5^9
(1, 11, 31, 32, 12)(2, 13, 33, 34, 14)(3, 15, 35, 36, 16)
(4, 17, 37, 38, 18) (5, 19, 39, 40, 20)(6, 21, 41, 42, 22)
(7, 23, 43, 44, 24)(8, 25, 45, 46, 26)(9, 27, 47, 48, 28)
(10, 29, 49, 50, 30)
(1, 2, 3, 4, 5)(6, 10, 9, 8, 7)(11, 14, 16, 17, 20)(12, 13, 15, 18, 19)
(21, 29, 27, 25, 24)(22, 30, 28, 26, 23)(31, 34, 36, 37, 40)
(32, 33, 35, 38, 39)(41, 49, 47, 45, 44)(42, 50, 48, 46, 43)
(1, 6)(2, 7)(3, 8)(4, 9)(5, 10)(11, 21, 12, 22)(13, 23, 14, 24)
(15, 26, 16, 25)(17, 27, 18, 28)(19, 30, 20, 29)(31, 41, 32, 42)
(33, 43, 34, 44)(35, 46, 36, 45)(37, 47, 38, 48)(39, 50, 40, 49)
> Ker;
Permutation group Ker acting on a set of cardinality 100
Order = 1
Id(Ker)
> // Thus G acts faithfully on this block system.
> 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) >;
> O1 := Orbits(G);
> O1;
[
GSet{ 1, 3, 6, 8, 9, 11, 12, 14, 15, 17, 24, 26, 27, 29,
30, 32, 33, 35, 38, 40, 41, 43, 46, 48 },
GSet{ 2, 4, 5, 7, 10, 13, 16, 18, 19, 20, 21, 22, 23, 25, 28,
31, 34, 36, 37, 39, 42, 44, 45, 47 }
]
> // Thus G has two orbits each of size 24
> f1, Im1, Ker1 := OrbitAction(G, O1[1]);
> FactoredOrder(Im1);
[ <2, 7>, <3, 9>, <5, 1>, <7, 1> ]
> IsPrimitive(Im1);
false
> P1 := MinimalPartition(Im1);
> P1;
GSet{
{ 11, 19, 21 },
{ 4, 8, 9 },
{ 3, 6, 7 },
{ 14, 15, 18 },
{ 2, 10, 24 },
{ 12, 13, 17 },
{ 16, 20, 22 },
{ 1, 5, 23 }
}
> f2, Im2, Ker2 := BlocksAction(Im1, P1);
> FactoredOrder(Im2);
[ <2, 7>, <3, 2>, <5, 1>, <7, 1> ]
> IsPrimitive(Im2);
true
> IsSymmetric(Im2);
true
> FactoredOrder(Ker2);
[ <3, 7> ]
> IsElementaryAbelian(Ker2);
true
> /*
> Hence the group obtained by restricting G to the first orbit consists
> of Sym(8) acting on an elementary abelian subgroup of order 3^7.
> */
> O2 := Orbits(Ker1);
> O2;
[
GSet{ 1 },
GSet{ 2, 7, 44, 16, 19, 20, 36, 10, 42, 22, 28, 21, 23, 25, 4,
39, 47, 34, 13, 45, 37, 31, 18, 5 },
GSet{ 3 },
GSet{ 6 },
GSet{ 8 },
GSet{ 9 },
GSet{ 11 },
GSet{ 12 },
GSet{ 14 },
GSet{ 15 },
GSet{ 17 },
GSet{ 24 },
GSet{ 26 },
GSet{ 27 },
GSet{ 29 },
GSet{ 30 },
GSet{ 32 },
GSet{ 33 },
GSet{ 35 },
GSet{ 38 },
GSet{ 40 },
GSet{ 41 },
GSet{ 43 },
GSet{ 46 },
GSet{ 48 }
]
> f3, Im3, Ker3 := OrbitAction(Ker1, O2[2]);
> FactoredOrder(Im3);
[ <2, 20>, <3, 5>, <5, 2>, <7, 1>, <11, 1> ]
> FactoredOrder(Ker3);
[]
> P := MinimalPartition(Im3);
> P;
GSet{
{ 1, 24 },
{ 2, 5 },
{ 3, 7 }
{ 3, 7 },
{ 4, 6 },
{ 22, 8 },
{ 9, 10 },
{ 11, 12 },
{ 23, 13 },
{ 14, 18 },
{ 15, 17 },
{ 16, 19 },
{ 20, 21 }
}
> f4, Im4, Ker4 := BlocksAction(Im3, P);
> Im4;
Permutation group Im4 acting on a set of cardinality 12
Order = 239500800 = 2^9 * 3^5 * 5^2 * 7 * 11
(10, 12, 11)
(9, 12, 11)
(8, 12, 9)
(7, 9)(8, 12)
(6, 9, 10)
(5, 6, 9)
(4, 6, 9)
(3, 6, 9)
(2, 6, 9, 5)(4, 10)
(1, 9, 6, 5)(4, 10)
> IsPrimitive(Im4);
true
> IsAlternating(Im4);
true
> FactoredOrder(Ker4);
[ <2, 11> ]
> IsElementaryAbelian(Ker4);
true
> /*
> The kernel of the restriction of G to the first orbit is isomomorphic
> to Alt(12) acting on an elementary abelian group of order 2^11.
> */
Given a primitive group G which has a non-trivial elementary abelian regular normal subgroup A, construct the representation of G given by the action of G on a basis for the elementary abelian group A. As with the other action functions, AffineAction returns the homomorphism, the image and the kernel of the action.
Given a primitive group G which has a non-trivial elementary abelian regular normal subgroup A, construct the permutation group that results from the action of G on a basis for the elementary abelian group A.
Given a primitive group G which has a non-trivial elementary abelian regular normal subgroup A, construct the kernel of the action of G on a basis for the elementary abelian group A.
Given a primitive group G which has a non-abelian socle N, construct the permutation representation of G given by the action of G on the simple factors of N. As with the other action functions, SocleAction returns the homomorphism, the image and the kernel of the action.
Given a primitive group G which has a non-abelian socle N, construct the permutation group L induced by the action of G on the simple factors of N.
Given a primitive group G which has a non-abelian socle N, construct the kernel of the action of G on the simple factors of N.
Right coset of the subgroup H of the group G, where g is an element of G.
The double coset H * g * K of the subgroups H and K of the group G, where g is an element of G.
True if element g of group G lies in the coset C.
True if element g of group G does not lie in the coset C.
True if the coset C_1 is equal to the coset C_2.
True if the coset C_1 is not equal to the coset C_2.
The cardinality of the coset C.
Set of double cosets H * g * K of the group G.
The (right) coset table for G over subgroup H relative to its defining generators.
The coset table for G corresponding to the permutation representation f of G, where f is a homomorphism of G onto a transitive permutation group.
Given a permutation group G and a subgroup H of G, this function returnsand the corresponding transversal mapping.
- An indexed set of elements T of G forming a right transversal for G over H; and
- The corresponding transversal mapping phi: G -> T. If T = [t_1, ..., t_r] and g in G, phi is defined by phi(g) = t_i, where g in H * t_i.
A system of coset representatives for the double cosets H * g * K in the group G, and the corresponding transversal mapping.
Given a subgroup H of the group G, construct the permutation representation of G given by the action of G on the (right) coset space cos(G, H). The function returns:Note that G may be any type of group. If G is a finitely presented group, then K may be returned undefined.
- The natural homomorphism f: G -> L;
- The induced group L;
- The kernel K of the action (a subgroup of G).
Given a subgroup H of the group G, construct the image L of G given by the action of G on the (right) coset space cos(G, H). L is returned as a permutation group.
Given a subgroup H of the group G, construct the kernel of the action of G on the (right) coset space cos(G, H).[Next] [Prev] [Right] [Left] [Up] [Index] [Root]