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A104404 Number of groups of order n all of whose subgroups are normal. 6
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 (list; graph; refs; listen; history; text; internal format)
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

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

J. C. Lennox, S. E. Stonehewer, Subnormal Subgroups of Groups, Oxford University Press, 1987.

LINKS

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

B. Horvat, G. Jaklic and T. Pisanski, On the number of Hamiltonian groups, 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

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. A000688, A000001, A104488.

Sequence in context: A300384 A252890 A173398 * A162512 A162510 A292589

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

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Last modified October 23 09:28 EDT 2019. Contains 328345 sequences. (Running on oeis4.)