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A037852
Number of normal subgroups of dihedral group with 2n elements.
2
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
1,1
COMMENTS
When n is an odd prime a(n) = 3.
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
FORMULA
a(n) = d(n) + 2 + (-1)^n. - Paul Boddington, Feb 02 2004
a(n) = A000005(n) + A176040(n). - Michel Marcus, Aug 19 2015
EXAMPLE
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
PROG
(PARI) a(n) = numdiv(n) + 2 + (-1)^n \\ Michel Marcus, Jul 30 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 04 2001
EXTENSIONS
More terms from Michel Marcus, Jul 30 2013
STATUS
approved