A037852 Number of normal subgroups of dihedral group with 2n elements.

2, 5, 3, 6, 3, 7, 3, 7, 4, 7, 3, 9, 3, 7, 5, 8, 3, 9, 3, 9, 5, 7, 3, 11, 4, 7, 5, 9, 3, 11, 3, 9, 5, 7, 5, 12, 3, 7, 5, 11, 3, 11, 3, 9, 7, 7, 3, 13, 4, 9, 5, 9, 3, 11, 5, 11, 5, 7, 3, 15, 3, 7, 7, 10, 5, 11, 3, 9, 5, 11, 3, 15, 3, 7, 7

Offset: 1

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 04 2001

Keywords

• nonn
• easy

Comments

When n is an odd prime a(n) = 3.

Write D_{2n} as , then the subgroups are of the form for d|n or for d|n and 0 <= r < d. The normal subgroups are for d|n and for d|gcd(n,2) and 0 <= r < d. There are d(n) normal subgroups of the first type and sigma(gcd(n,2)) normal subgroups of the second type. - Jianing Song, Jul 21 2022

Examples

			a(4) = 6 since D_8 = <a, x | a^4 = x^2 = 1, x*a*x = a^(-1)> has 6 normal subgroups: {e}, {e,a^2}, {e,a,a^2,a^3}, {e,a^2,x,a^2*x}, {e,a^2,a*x,a^3*x} and D_8. The 4 subgroups {e,x}, {e,a*x}, {e,a^2*x} and {e,a^3*x} are not normal. - _Jianing Song_, Jul 21 2022
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1001
• Keith Conrad, Dihedral Groups II
• The Group Properties Wiki, Subgroup structure of dihedral groups
• Index entries for sequences related to groups

Crossrefs

Cf. A062249, A007503, A062553.

Cf. A000005, A176040.

Programs

PARI

a(n) = numdiv(n) + 2 + (-1)^n \\ Michel Marcus, Jul 30 2013

Formula

a(n) = d(n) + 2 + (-1)^n. - Paul Boddington, Feb 02 2004

a(n) = A000005(n) + A176040(n). - Michel Marcus, Aug 19 2015

Extensions

More terms from Michel Marcus, Jul 30 2013