Finite Permutation Groups¶
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class
sage.categories.finite_permutation_groups.
FinitePermutationGroups
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom
The category of finite permutation groups, i.e. groups concretely represented as groups of permutations acting on a finite set.
It is currently assumed that any finite permutation group comes endowed with a distinguished finite set of generators (method
group_generators
); this is the case for all the existing implementations in Sage.EXAMPLES:
sage: C = PermutationGroups().Finite(); C Category of finite permutation groups sage: C.super_categories() [Category of permutation groups, Category of finite groups, Category of finite finitely generated semigroups] sage: C.example() Dihedral group of order 6 as a permutation group
TESTS:
sage: C is FinitePermutationGroups() True sage: TestSuite(C).run() sage: G = FinitePermutationGroups().example() sage: TestSuite(G).run(verbose = True) running ._test_an_element() . . . pass running ._test_associativity() . . . pass running ._test_cardinality() . . . pass running ._test_category() . . . pass running ._test_elements() . . . Running the test suite of self.an_element() running ._test_category() . . . pass running ._test_eq() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_pickling() . . . pass pass running ._test_elements_eq_reflexive() . . . pass running ._test_elements_eq_symmetric() . . . pass running ._test_elements_eq_transitive() . . . pass running ._test_elements_neq() . . . pass running ._test_enumerated_set_contains() . . . pass running ._test_enumerated_set_iter_cardinality() . . . pass running ._test_enumerated_set_iter_list() . . . pass running ._test_eq() . . . pass running ._test_inverse() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_one() . . . pass running ._test_pickling() . . . pass running ._test_prod() . . . pass running ._test_some_elements() . . . pass
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class
ElementMethods
¶
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class
FinitePermutationGroups.
ParentMethods
¶ -
cycle_index
(parent=None)¶ INPUT:
self
- a permutation group \(G\)parent
– a free module with basis indexed by partitions, or behave as such, with aterm
andsum
method (default: the symmetric functions over the rational field in the p basis)
Returns the cycle index of \(G\), which is a gadget counting the elements of \(G\) by cycle type, averaged over the group:
\[P = \frac{1}{|G|} \sum_{g\in G} p_{ \operatorname{cycle\ type}(g) }\]EXAMPLES:
Among the permutations of the symmetric group \(S_4\), there is the identity, 6 cycles of length 2, 3 products of two cycles of length 2, 8 cycles of length 3, and 6 cycles of length 4:
sage: S4 = SymmetricGroup(4) sage: P = S4.cycle_index() sage: 24 * P p[1, 1, 1, 1] + 6*p[2, 1, 1] + 3*p[2, 2] + 8*p[3, 1] + 6*p[4]
If \(l = (l_1,\dots,l_k)\) is a partition,
|G| P[l]
is the number of elements of \(G\) with cycles of length \((p_1,\dots,p_k)\):sage: 24 * P[ Partition([3,1]) ] 8
The cycle index plays an important role in the enumeration of objects modulo the action of a group (Polya enumeration), via the use of symmetric functions and plethysms. It is therefore encoded as a symmetric function, expressed in the powersum basis:
sage: P.parent() Symmetric Functions over Rational Field in the powersum basis
This symmetric function can have some nice properties; for example, for the symmetric group \(S_n\), we get the complete symmetric function \(h_n\):
sage: S = SymmetricFunctions(QQ); h = S.h() sage: h( P ) h[4]
TODO: add some simple examples of Polya enumeration, once it will be easy to expand symmetric functions on any alphabet.
Here are the cycle indices of some permutation groups:
sage: 6 * CyclicPermutationGroup(6).cycle_index() p[1, 1, 1, 1, 1, 1] + p[2, 2, 2] + 2*p[3, 3] + 2*p[6] sage: 60 * AlternatingGroup(5).cycle_index() p[1, 1, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 24*p[5] sage: for G in TransitiveGroups(5): # optional - database_gap # long time ... G.cardinality() * G.cycle_index() p[1, 1, 1, 1, 1] + 4*p[5] p[1, 1, 1, 1, 1] + 5*p[2, 2, 1] + 4*p[5] p[1, 1, 1, 1, 1] + 5*p[2, 2, 1] + 10*p[4, 1] + 4*p[5] p[1, 1, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 24*p[5] p[1, 1, 1, 1, 1] + 10*p[2, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 20*p[3, 2] + 30*p[4, 1] + 24*p[5]
One may specify another parent for the result:
sage: F = CombinatorialFreeModule(QQ, Partitions()) sage: P = CyclicPermutationGroup(6).cycle_index(parent = F) sage: 6 * P B[[1, 1, 1, 1, 1, 1]] + B[[2, 2, 2]] + 2*B[[3, 3]] + 2*B[[6]] sage: P.parent() is F True
This parent should have a
term
andsum
method:sage: CyclicPermutationGroup(6).cycle_index(parent = QQ) Traceback (most recent call last): ... AssertionError: `parent` should be (or behave as) a free module with basis indexed by partitions
REFERENCES:
[Ker1991] A. Kerber. Algebraic combinatorics via finite group actions, 2.2 p. 70. BI-Wissenschaftsverlag, Mannheim, 1991. AUTHORS:
- Nicolas Borie and Nicolas M. Thiery
TESTS:
sage: P = PermutationGroup([]); P Permutation Group with generators [()] sage: P.cycle_index() p[1] sage: P = PermutationGroup([[(1)]]); P Permutation Group with generators [()] sage: P.cycle_index() p[1]
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FinitePermutationGroups.
example
()¶ Returns an example of finite permutation group, as per
Category.example()
.EXAMPLES:
sage: G = FinitePermutationGroups().example(); G Dihedral group of order 6 as a permutation group
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FinitePermutationGroups.
extra_super_categories
()¶ Any permutation group is assumed to be endowed with a finite set of generators.
TESTS:
sage: PermutationGroups().Finite().extra_super_categories() [Category of finitely generated magmas]
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class