A384065 Cardinality of the lattice of order ideals for every order ideal in the lattice of normal subgroups of the dihedral group D_{2*n}.

3, 10, 4, 11, 4, 16, 4, 12, 5, 16, 4, 20, 4, 16, 7, 13, 4, 23, 4, 20, 7, 16, 4, 25, 5, 16, 6, 20, 4, 39, 4, 14, 7, 16, 7, 33, 4, 16, 7, 25, 4, 39, 4, 20, 11, 16, 4, 31, 5, 23, 7, 20, 4, 31, 7, 25, 7, 16, 4, 69, 4, 16, 11, 15, 7, 39, 4, 20, 7, 39, 4, 48, 4, 16, 11, 20, 7, 39

Offset: 1

Author

Miles Englezou, May 18 2025

Keywords

• nonn

Comments

An order ideal I is a nonempty subset of a partially ordered set P where y <= x for x in I and y in P implies y is in I. The set of order ideals of a partially ordered set is a lattice (ordered by inclusion) and is distributive.

The lattice of normal subgroups of any group is distributive and also modular.

The number of order ideals in the lattice of normal subgroups of D_{2*n} is a(n)-1 since the empty set is not an order ideal.

Examples

			a(1) = 3 since for D_{2} = C_2 the lattice of normal subgroups L = {1, C_2} and the lattice of order ideals of L contains {}, {1}, and L.
a(2) = 10 since for D_{4} = (C_2 x C_2) the lattice of normal subgroups L = {1, C_2, C_2, C_2, (C_2 x C_2)} and the lattice of order ideals of L contains {}, {1}, {1, C_2}, {1, C_2}, {1, C_2}, {1, C_2, C_2}, {1, C_2, C_2}, {1, C_2, C_2}, {1, C_2, C_2, C_2}, and L.
		

Links

• Wikipedia, Ideal (order theory)

Crossrefs

Cf. A037852.

Programs

Sage

def a(n):
    return len(Poset((DihedralGroup(n).normal_subgroups(), lambda H, K: H.is_subgroup(K))).order_ideals_lattice())

Formula

a(n) > A037852(n).

a(p) = 4 for every prime p > 2.