Given an integer n ≥1, create the generic permutation group acting on the natural G-set Ω = { 1, 2, ..., n }, i.e. the symmetric group Sym(Ω). Initially, only a structure table is created for Sym(n), so that, in particular, generators are not defined. This function is normally used to provide a context for the creation of elements and subgroups of Sym(n). If structural computation is attempted with the group created by Sym(n), then generators will be created dynamically.
Given a finite set X of cardinality n ≥1, create the generic group G of permutations of X -- the symmetric group Sym(X). Initially, only a structure table is created for Sym(X), so that, in particular, generators are not defined. This function is normally used to provide a context for the creation of elements and subgroups of Sym(X). If structural computation is attempted with the group created by Sym(X), then generators will be created dynamically. Although the group G is defined on the set X, G is represented internally as a group of permutations of the set Ω = { 1, 2, ..., n }. Translation between X and Ω is done at input/output time. The precise representation can be found by using the Labelling function. If X is an indexed set then the indexing of elements of X determines the correspondence.
Return a group H isomorphic to G, but acting on the standard set { 1, ..., n }. This function is useful when the natural G-set for G is not the standard set. If the natural G-set for G is the standard set, G is returned. The isomorphism from G to H is also returned.
> S4 := Sym({ "a", "b", "c", "d" });
> S4;
Symmetric group S4 acting on a set of cardinality 4
Order = 24 = 2^3 * 3
> GSet(S4);
GSet{ at c, b, a, d atbrace
We define the symmetric group of degree 10 acting on the
set { 0, 1, ..., 9}.
> G := Sym({ 0..9 });
> G;
Symmetric group G acting on a set of cardinality 10
Order = 3628800 = 2^8 * 3^4 * 5^2 * 7
> GSet(G);
GSet{ at 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 atbrace
Throughout this subsection we shall assume that the permutation group G has natural G-set X.
Given a permutation group G defined as acting on the set X = { x1, ..., xn} of cardinality n ≥1, and a list L of distinct elements a1, a2, ..., an of X, construct the element g of G defined by xi -> ai, for i = 1, ..., n. Unless G is known to be the generic permutation group of degree n, the permutation will be tested for membership of G, and if g is not an element of G, the function will fail. If g does lie in G, g will have G as its parent. Since the membership test may involve constructing a base and strong generating set for G, this constructor may occasionally be very costly. Hence, a permutation g should be defined as an element of a subgroup of the generic group only when membership of G is required by subsequent operations involving g.
Given a permutation group G defined as acting on the set X = { x1, ..., xn} of cardinality n ≥1, and a sequence Q = [a1, a2, ..., an] of distinct elements of X, construct the permutation g of X defined by xi -> ai, for i = 1, ..., n. This permutation will have G as its parent structure. As in the case of the elt-constructor, the operation will fail if g is not an element of G and the same observations concerning the cost of membership testing apply.
Given a permutation group G defined as acting on the set X={x1, x2, ..., xn}, construct the permutation g corresponding to the given product of cycles. Adjacent letters must be separated by commas. Further, cycles of length one must be omitted. The coercion operator ! may be omitted only within the context of the standard constructors sub<>, ncl<> and quo<>. Once the permutation g has been constructed, it will be tested for membership in G. If it is not a member, the construction fails.
Given a permutation group G defined as acting on the set X={1 .. n}, construct the permutation g corresponding to the given product of literal cycles of integers. Adjacent integers must be separated by commas. Once the permutation g has been constructed, it will be tested for membership in G. If it is not a member, the construction fails. This construction is strongly recommended when creating large permutations to avoid overhead in constructing unnecessarily large parse trees by Magma.
Given a permutation group G defined as acting on the set X={1 .. n}, construct the permutation g corresponding to the given product of cycles. The indexed sets in Q must be disjoint subsets of X, which are interpreted as the disjoint cycles of the permutation being constructed. Cycles of length 1 may be omitted, but do not have to be omitted. Once the permutation g has been constructed, it will be tested for membership in G. If it is not a member, the construction fails. Note that the Cycle function produces results suitable for use as members of Q.
The sequence Q of images of the G-set of g. In particular, it has the property that Parent(g)!Eltseq(g) eq g.
Construct the identity permutation in the permutation group G.
> S6 := Sym(6); > x := elt<S6 | 1,3,2,5,6,4>; > x; (2, 3)(4, 5, 6) > y := S6![1,3,2,5,6,4]; > y; (2, 3)(4, 5, 6) > z := S6!(2,3)(4,5,6); > z; (2, 3)(4, 5, 6) > S6!1; Id(S6)
Suppose that the cardinality of the set X is n. Construct the permutation group G acting on the set X generated by the permutations defined by the list L. A term of the list L must be an object of one of the following types:Each element or group specified by the list must belong to the same generic permutation group. The group G will be constructed as a subgroup of some group which contains each of the elements and groups specified in the list.
- (a)
- A sequence of n elements of X defining a permutation of X (note that this is only well-defined when X is an indexed set);
- (b)
- A set or sequence of sequences of type (a);
- (c)
- An element of Sym(X);
- (d)
- A set or sequence of elements of Sym(X);
- (e)
- A subgroup of Sym(X);
- (f)
- A set or sequence of subgroups of Sym(X).
The generators of G consist of the elements specified by the terms of the list L together with the stored generators for groups specified by terms of the list.
The PermutationGroup constructor is shorthand for the two statements:
SX := Sym(X); G := sub< SX | L >;
where sub< ... > is the subgroup constructor described in the next subsection.
Construct the permutation group G acting on the set X = { 1, 2, ..., n } generated by the permutations defined by the list L. The possibilities for the terms of the list L are the same as for the constructor PermutationGroup< X | L >.
> H := PermutationGroup< 9 | (1,2,4)(5,6,8)(3,9,7), (4,5,6)(7,9,8) >;
> H;
Permutation group H acting on a set of cardinality 9
(1, 2, 4)(3, 9, 7)(5, 6, 8)
(4, 5, 6)(7, 9, 8)