, viewing g as a permutation matrix.Cat::PermutationGroup – the category of permutation groups
Cat::PermutationGroup represents the category of permutation groups, that is the category of groups of permutations of some finite set, or the category of finite groups acting faithfully on finite sets.
Creating the Category
Cat::PermutationGroup(n)
Parameters:
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n: |
A non negative integer |
Categories
Details:
A Cat::PermutationGroup is a.
Entries
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"allElements" |
returns the list of all the elements of the group. |
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"size" |
returns the size of the group. |
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"generators" |
returns a list or set of generators of the group. |
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"set" |
returns the finite set on which this permutation group acts.If it's a list, this defines the enumeration e1, e2, ..., en of the elements of the set S. If it is a set, this enumeration will be defined by MuPAD's internal order. |
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"invertSet" |
returns a table containing for each element of the set its index in the enumeration of the set. Therefore, if e is an element of the set, the expression op(dom::set, dom::invertSet[e]) will evaluate back to e. |
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"card" |
returns the size of the set on which this permutation group acts. |
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"orbits" |
returns the permutation in cycle representation (list of its cycles). |
actOnList – action on lists.
Cat::PermutationGroup::actOnList(dom g, list l)
Returns the result of the action of g on lists representing objects whose content is indexed by the set S on which this permutation group acts.
orbit – ...
Cat::PermutationGroup::orbit(list l)
Returns the ...
orbitSet – ...
Cat::PermutationGroup::orbitSet(list l)
Returns the ...
orbitSize – ...
Cat::PermutationGroup::orbitSize(list l)
Returns the ...
canonicCompositions – ...
Cat::PermutationGroup::canonicCompositions(nonnegative integer m)
Returns the ...
cycleType – cycle type of an element of the group.
Cat::PermutationGroup::cycleType(dom g)
Returns the cycle type of an element of the group. This is a list [k1, k2, ..., km] where m is the size of dom::set. This list indicates that there are ki cycles of length i.
charPolynomial – characteristic polynomial of a permutation
Cat::PermutationGroup::charPolynomial(dom g, variable z)
Returns the characteristic polynomial of the permutation g, that is, the determinant of the matrix , viewing g as a permutation matrix.
cycleTypes – count the elements of the group by cycle type.
Cat::PermutationGroup::cycleTypes()
Returns a table associating a cycle type to the number of elements of the group having this cycle type.
cycleIndicatorPoly – ...
Cat::PermutationGroup::cycleIndicatorPoly(DOM_IDENT z)
Returns the cycle indicator polynomial of the group, as a polynomial in 
PolyaSubs – Polya substitution.
Cat::PermutationGroup::PolyaSubs(P, weights)
...