A104404 -id:A104404 - cp's OEIS frontend

A104488 Number of Hamiltonian groups of order n.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

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

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.

T. D. Noe, 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.
Tomaž Pisanski and Thomas W. Tucker, The genus of low rank hamiltonian groups, Discrete Math. 78 (1989), 157-167.
Eric Weisstein's World of Mathematics, Hamiltonian Group.

Cf. A000265, A000688, A021002, A048651, A104404, A104404, A104452, A104453.

orders[n_]:=Map[Last, FactorInteger[n]]; a[n_]:=Apply[Times, Map[PartitionsP, orders[n]]]; e[n_]:=n/ 2^IntegerExponent[n, 2]; h[n_]/;Mod[n, 8]==0:=a[e[n]]; h[n_]:=0;
(* Second program: *)
a[n_] := If[Mod[n, 8]==0, FiniteAbelianGroupCount[n/2^IntegerExponent[n, 2]], 0]; Array[a, 102] (* Jean-François Alcover, Sep 14 2019 *)

a(n)={my(e=valuation(n, 2)); if(e<3, 0, my(f=factor(n/2^e)[, 2]); prod(i=1, #f, numbpart(f[i])))} \\ Andrew Howroyd, Aug 08 2018

Let n = 2^e*o, where e = e(n) >= 0 and o = o(n) is an odd number. The number h(n) of Hamiltonian groups of order n is given by h(n) = 0, if e(n) < 3 and h(n) = a(o(n)), otherwise, where a(n) = A000688(n) denotes the number of Abelian groups of order n.
a(8*n) = A000688(A000265(n)), a(n) = 0 for n mod 8 <> 0. - Andrew Howroyd, Aug 08 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A021002 * A048651 / 4 = 0.16568181590156732257... . - Amiram Eldar, Sep 23 2023

A104452 Number of groups of order <= n all of whose subgroups are normal.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 12, 14, 15, 16, 18, 19, 20, 21, 27, 28, 30, 31, 33, 34, 35, 36, 40, 42, 43, 46, 48, 49, 50, 51, 59, 60, 61, 62, 66, 67, 68, 69, 73, 74, 75, 76, 78, 80, 81, 82, 88, 90, 92, 93, 95, 96, 99, 100, 104, 105, 106, 107, 109, 110, 111, 113, 125, 126, 127

Offset: 1

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

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.

Amiram Eldar, 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.
Tomaž Pisanski and Thomas W. Tucker, The genus of low rank hamiltonian groups, Discrete Math. 78 (1989), 157-167.
Eric Weisstein's World of Mathematics, Abelian Group.
Eric Weisstein's World of Mathematics, Hamiltonian Group.

Partial sums of A104404.

Cf. A000688, A021002, A048651, A063966, A104488, A104407, A104453.

orders[n_]:=Map[Last, FactorInteger[n]]; a[n_]:=Apply[Times, Map[PartitionsP, orders[n]]]; e[n_]:=n/ 2^IntegerExponent[n, 2]; h[n_]/;Mod[n, 8]==0:=a[e[n]]; h[n_]:=0; numberOfAbelianGroupsOfOrderLEQThanN[n_]:=Map[Apply[Plus, # ]&, Table[Take[Map[a, Table[i, {i, 1, n}]], i], {i, 1, n}]]; numberOfHamiltonianGroupsOfOrderLEQThanN[n_]:=Map[Apply[Plus, # ]&, Table[Take[Map[h, Table[i, {i, 1, n}]], i], {i, 1, n}]]; numberOfAllGroupsOfOrderLEQThanN[n_]:=numberOfAbelianGroupsOfOrderLEQThanN[n] +numberOfHamiltonianGroupsOfOrderLEQThanN[n];

a(n) ~ c * n, where c = A021002 * (1 + A048651/4) = 2.46053840757488111675... . - Amiram Eldar, Oct 03 2023

A104407 Number of Hamiltonian groups of order <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12

Offset: 1

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

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.

Amiram Eldar, 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.
Tomaž Pisanski and Thomas W. Tucker, The genus of low rank hamiltonian groups, Discrete Math. 78 (1989), 157-167.
Eric Weisstein's World of Mathematics, Hamiltonian Group.

Partial sums of A104488.

Cf. A000688, A021002, A048651, A063966, A104404, A104452, A104453.

orders[n_]:=Map[Last, FactorInteger[n]]; a[n_]:=Apply[Times, Map[PartitionsP, orders[n]]]; e[n_]:=n/ 2^IntegerExponent[n, 2]; h[n_]/;Mod[n, 8]==0:=a[e[n]]; h[n_]:=0; numberOfHamiltonianGroupsOfOrderLEQThanN[n_]:=Map[Apply[Plus, # ]&, Table[Take[Map[h, Table[i, {i, 1, n}]], i], {i, 1, n}]];

a(n) ~ c * n, where c = A021002 * A048651 / 4 = 0.16568181590156732257... . - Amiram Eldar, Oct 03 2023

A104453 Smallest order for which there are n nonisomorphic finite Hamiltonian groups, or 0 if no such order exists.

Original entry on oeis.org

8, 72, 216, 1800, 648, 5400, 1944, 88200, 27000, 16200, 10, 5832, 264600, 0, 48600, 17496, 10672200, 0, 1323000, 0, 793800, 20, 243000, 52488, 0, 32016600, 405000, 0, 9261000, 2381400, 0, 157464

Offset: 1

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

R. D. Carmichael, Introduction to the Theory of Groups of Finite Order, New York, Dover, 1956.
J. C. Lennox and S. E. Stonehewer, Subnormal Subgroups of Groups, Oxford University Press, 1987.

B. Horvat, G. Jaklic and T. Pisanski, On the number of Hamiltonian groups, arXiv:math/0503183 [math.CO], 2005.
T. Pisanski and T.W. Tucker, The genus of low rank hamiltonian groups, Discrete Math. 78 (1989), 157-167.
Eric Weisstein's World of Mathematics, Abelian Group
Eric Weisstein's World of Mathematics, Hamiltonian Group

Cf. A000688, A063966, A104488, A104407, A104404, A104452.

S_h(n) denotes the smallest number k for which exactly n nonisomorphic hamiltonian groups of order k exist. Here 0 indicates the case when n is not a product of partition numbers and S_h(n) does not exist.

A235388 Number of groups of order 2n generated by involutions.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 4, 1, 1, 1, 12, 1, 3, 1, 3, 1, 1, 1, 11, 2, 1, 4, 3, 1, 3, 1, 49, 1, 1, 1, 12, 1, 1, 1, 9, 1, 2, 1, 3, 2, 1, 1, 46, 2, 3, 1, 3, 1, 8, 1, 9, 1, 1, 1, 10, 1, 1, 2, 359, 1, 2, 1, 3, 1, 2, 1, 40, 1, 1, 3, 3, 1, 2, 1, 38, 11, 1, 1

Offset: 1

Eric M. Schmidt, Jan 08 2014

nonn

a(n) >= A104404(n). This can be proved using the characterization in A104404. Given an Abelian group G, the semidirect product G : , where h^2 = 1 and hgh = g^(-1) for any g in G, is generated by involutions. There is also a semidirect product Q8 : C2 generated by involutions. So an involution-generated group G : C2 exists for any finite group G that has all subgroups normal, and it can be shown that they are all nonisomorphic.

Eric M. Schmidt, Table of n, a(n) for n = 1..511

IsInvolutionGenerated := G -> Group(Filtered(G, g->g^2=Identity(G)))=G;
A235388 := function(n) local i, count; count := 0; for i in [1..NrSmallGroups(2*n)] do if IsInvolutionGenerated(SmallGroup(2*n, i)) then count := count + 1; fi; od; return count; end;

cp's OEIS Frontend

A104488 Number of Hamiltonian groups of order n.

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A104452 Number of groups of order <= n all of whose subgroups are normal.

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A104407 Number of Hamiltonian groups of order <= n.

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A104453 Smallest order for which there are n nonisomorphic finite Hamiltonian groups, or 0 if no such order exists.

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A235388 Number of groups of order 2n generated by involutions.

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