public Group <Permutation <T> > InnerAutomorphismGroup()
{
return(new Group <Permutation <T> >(
new FiniteSet <Permutation <T> >(_set.Select(g => Permutation <T> .FromFunction(_set, x => _product(_inverse(g), _product(x, g))))),
Permutation <T> .Product,
Permutation <T> .Inverse,
Permutation <T> .Identity(_set)
));
}
public Group <Permutation <U> > ActionGroup <U>(Func <T, U, U> action, FiniteSet <U> set) where U : IEquatable <U>
{
return(new Group <Permutation <U> >(
new FiniteSet <Permutation <U> >(_set.Select(g => Permutation <U> .FromFunction(set, x => action(g, x)))),
Permutation <U> .Product,
Permutation <U> .Inverse,
Permutation <U> .Identity(set)
));
}
public Group <Permutation <T> > AutomorphismGroup()
{
// TODO: Optimize.
return(new Group <Permutation <T> >(
new FiniteSet <Permutation <T> >(Permutation <T> .SetPermutations(_set).Where(permutation => this.IsAutomorphism(permutation.Evaluate))),
Permutation <T> .Product,
Permutation <T> .Inverse,
Permutation <T> .Identity(_set)
));
}
/// <summary>
/// Returns the cyclic group on the vertices {1,2,...,n} of an n-gon.
/// </summary>
public static Group <Permutation <int> > CyclicGroupP(int n)
{
FiniteSet <int> set = new FiniteSet <int>(Enumerable.Range(1, n));
HashSet <Permutation <int> > permutations = new HashSet <Permutation <int> >();
Permutation <int> generator = Permutation <int> .FromCycle(set, new Cycle <int>(Enumerable.Range(1, n)));
Permutation <int> current = Permutation <int> .Identity(set);
for (int i = 0; i < n; i++)
{
permutations.Add(current);
current = Permutation <int> .Product(current, generator);
}
return(new Group <Permutation <int> >(new FiniteSet <Permutation <int> >(permutations), Permutation <int> .Product, Permutation <int> .Inverse, Permutation <int> .Identity(set)));
}
/// <summary>
/// Returns the alternating group Alt(S) of order |S|!/2.
/// </summary>
public static Group <Permutation <T> > AlternatingGroup <T>(FiniteSet <T> set) where T : IEquatable <T>
{
FiniteSet <Permutation <T> > permutations = new FiniteSet <Permutation <T> >(Permutation <T> .SetPermutations(set).Where(a => a.Sign == 1));
return(new Group <Permutation <T> >(permutations, Permutation <T> .Product, Permutation <T> .Inverse, Permutation <T> .Identity(set)));
}
/// <summary>
/// Returns the symmetric group Sym(S) of order |S|!.
/// </summary>
public static Group <Permutation <T> > SymmetricGroup <T>(FiniteSet <T> set) where T : IEquatable <T>
{
FiniteSet <Permutation <T> > permutations = new FiniteSet <Permutation <T> >(Permutation <T> .SetPermutations(set));
return(new Group <Permutation <T> >(permutations, Permutation <T> .Product, Permutation <T> .Inverse, Permutation <T> .Identity(set)));
}
/// <summary>
/// Returns a permutation group on {1,2,...,|G|} isomorphic to this group G.
/// </summary>
/// <param name="bijection">A bijection from G to {1,2,...,|G|}.</param>
/// <param name="bijectionInverse">The inverse of <paramref name="bijection"/>.</param>
/// <param name="isomorphism">An isomorphism from G to the permutation group.</param>
public Group <Permutation <int> > ToPermutationGroup(Func <T, int> bijection, Func <int, T> bijectionInverse, out Func <T, Permutation <int> > isomorphism)
{
if (bijectionInverse == null)
{
bijectionInverse = Discrete.Inverse(_set, bijection);
}
var set = new FiniteSet <int>(Enumerable.Range(1, _set.Size));
isomorphism = x => Permutation <int> .FromFunction(set, k => bijection(_product(x, bijectionInverse(k))));
var permutations = new FiniteSet <Permutation <int> >(_set.Select(isomorphism));
return(new Group <Permutation <int> >(permutations, Permutation <int> .Product, Permutation <int> .Inverse, Permutation <int> .Identity(set)));
}