cp's OEIS Frontend

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

%I A037852 #26 Jul 25 2025 09:01:00
%S A037852 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,
%T A037852 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,
%U A037852 10,5,11,3,9,5,11,3,15,3,7,7
%N A037852 Number of normal subgroups of dihedral group with 2n elements.
%C A037852 When n is an odd prime a(n) = 3.
%C A037852 Write D_{2n} as <a, x | a^n = x^2 = 1, x*a*x = a^(-1)>, then the subgroups are of the form <a^d> for d|n or <a^d, a^r*x> for d|n and 0 <= r < d. The normal subgroups are <a^d> for d|n and <a^d, a^r*x> 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
%H A037852 Antti Karttunen, <a href="/A037852/b037852.txt">Table of n, a(n) for n = 1..1001</a>
%H A037852 Keith Conrad, <a href="https://kconrad.math.uconn.edu/blurbs/grouptheory/dihedral2.pdf">Dihedral Groups II</a>
%H A037852 The Group Properties Wiki, <a href="https://groupprops.subwiki.org/wiki/Subgroup_structure_of_dihedral_groups">Subgroup structure of dihedral groups</a>
%H A037852 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%F A037852 a(n) = d(n) + 2 + (-1)^n. - _Paul Boddington_, Feb 02 2004
%F A037852 a(n) = A000005(n) + A176040(n). - _Michel Marcus_, Aug 19 2015
%e A037852 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
%o A037852 (PARI) a(n) = numdiv(n) + 2 + (-1)^n \\ _Michel Marcus_, Jul 30 2013
%Y A037852 Cf. A062249, A007503, A062553.
%Y A037852 Cf. A000005, A176040.
%K A037852 nonn,easy
%O A037852 1,1
%A A037852 Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 04 2001
%E A037852 More terms from _Michel Marcus_, Jul 30 2013