A104404 Number of groups of order n all of whose subgroups are normal.

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 6, 2, 2, 1, 2, 1, 3, 1, 4, 1, 1, 1, 2, 1, 1, 2, 12, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 6, 5, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 8, 1, 2, 2, 4, 1, 1

Offset

1,4

Comments

A finite non-Abelian group has all of its subgroups normal precisely when it is the direct product of the quaternion group of order 8, a (possibly trivial) elementary Abelian 2-group, and an Abelian group of odd order. [Carmichael, p. 114] - Eric M. Schmidt, Jan 12 2014

References

Robert D. Carmichael, Introduction to the Theory of Groups of Finite Order, New York, Dover, 1956.

John C. Lennox and Stewart. E. Stonehewer, Subnormal Subgroups of Groups, Oxford University Press, 1987.

Links

Hans Havermann, Table of n, a(n) for n = 1..10000

Boris Horvat, Gašper Jaklič, and Tomaž Pisanski, On the number of hamiltonian groups, Mathematical Communications, Vol. 10, No. 1 (2005), pp. 89-94; arXiv preprint, arXiv:math/0503183 [math.CO], 2005.

Eric Weisstein's World of Mathematics, Abelian Group.

Eric Weisstein's World of Mathematics, Hamiltonian Group.

Formula

The number a(n) of all groups of order n all of whose subgroups are normal is given as a(n) = b(n) + h(n), where b(n) denotes the number of Abelian groups of order n and h(n) denotes the number of Hamiltonian groups of order n.

a(n) = A000688(n) + A104488(n). - Andrew Howroyd, Aug 08 2018

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A021002 * (1 + A048651/4) = 2.46053840757488111675... . - Amiram Eldar, Sep 23 2023

Mathematica

orders[n_]:=Map[Last, FactorInteger[n]]; b[n_]:=Apply[Times, Map[PartitionsP, orders[n]]]; e[n_]:=n/ 2^IntegerExponent[n, 2]; h[n_]/; Mod[n, 8]==0:=b[e[n]]; h[n_]:=0; a[n_]:= b[n]+h[n];

Prog

(PARI) a(n)={my(e=valuation(n, 2)); my(f=factor(n/2^e)[, 2]); prod(i=1, #f, numbpart(f[i]))*(numbpart(e) + (e>=3))} \\ Andrew Howroyd, Aug 08 2018

Crossrefs

Cf. A000001, A000688, A021002, A048651, A104488.

Sequence in context: A370771 A368106 A368104 * A351655 A351656 A162512

Adjacent sequences: A104401 A104402 A104403 * A104405 A104406 A104407

Keyword

nonn,easy,mult

Author

Boris Horvat (Boris.Horvat(AT)fmf.uni-lj.si), Gasper Jaklic (Gasper.Jaklic(AT)fmf.uni-lj.si), Tomaz Pisanski, Apr 19 2005

Extensions

Keyword:mult added by Andrew Howroyd, Aug 08 2018

Status

approved