/// <summary>
/// Returns the semidirect product N x_f H.
/// </summary>
public static Group <Pair <T, U> > SemidirectProduct <T, U>(Group <T> n, Group <U> h, Func <U, T, T> f)
where T : IEquatable <T>
where U : IEquatable <U>
{
return(new Group <Pair <T, U> >(
FiniteSet.CartesianProduct(n.Set, h.Set),
(g1, g2) => new Pair <T, U>(n.Product(g1.First, f(g1.Second, g2.First)), h.Product(g1.Second, g2.Second)),
g => { U inverse = h.Inverse(g.Second); return new Pair <T, U>(f(inverse, n.Inverse(g.First)), inverse); },
new Pair <T, U>(n.Identity, h.Identity)
));
}
/// <summary>
/// Returns the direct product G_1 x G_2.
/// </summary>
public static Group <Pair <T, U> > DirectProduct <T, U>(Group <T> group1, Group <U> group2)
where T : IEquatable <T>
where U : IEquatable <U>
{
return(new Group <Pair <T, U> >(
FiniteSet.CartesianProduct(group1.Set, group2.Set),
(g1, g2) => new Pair <T, U>(group1.Product(g1.First, g2.First), group2.Product(g1.Second, g2.Second)),
g => new Pair <T, U>(group1.Inverse(g.First), group2.Inverse(g.Second)),
new Pair <T, U>(group1.Identity, group2.Identity)
));
}