A090091 Number of groups of order 3^n.

1, 1, 2, 5, 15, 67, 504, 9310, 1396077, 5937876645

Offset

0,3

References

G. Bagnera, La composizione dei Gruppi finiti il cui grado e la quinta potenza di un numero primo, Ann. Mat. Pura Appl. (3), 1 (1898), 137-228.

Hans Ulrich Besche, Bettina Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, International Journal of Algebra and Computation, Vol. 12, No 5 (2002), 623-644.

W. Burnside, Theory of Groups of Finite Order, Dover, NY, 1955.

Marcus du Sautoy, Symmetry: A Journey into the Patterns of Nature, HarperCollins, 2008, p. 96.

Links

Table of n, a(n) for n=0..9.

David Burrell, The number of p-groups of order 19,683 and new lists of p-groups, Communications in Algebra, Vol. 51 - Issue 6 (2023), 2673-2679.

Heiko Dietrich, Computational aspects of finite p-groups, 2016.

Rodney James and John Cannon, Computation of isomorphism classes of p-groups, Mathematics of Computation 23.105 (1969): 135-140.

M. F. Newman, E. A. O'Brien and M. R. Vaughan-Lee, Groups and nilpotent Lie rings whose order is the sixth power of a prime, J. Algebra, 278 (2004), 383-401.

E. A. O'Brien and M. R. Vaughan-Lee, The groups of order p^7 for odd prime p, J. Algebra 292, 243-258, 2005. [David Radcliffe, Feb 24 2010]

Michael Vaughan-Lee, Graham Higman’s PORC Conjecture, Jahresbericht der Deutschen Mathematiker-Vereinigung Vol. 114 (2012), 89-16.

Michael Vaughan-Lee, Groups of order p^8 and exponent p, International Journal of Group Theory Vol. 4 (2015), 25-42.

Brett Edward Witty, Enumeration of groups of prime-power order, PhD thesis, 2006.

Formula

a(n) = A000001(3^n).

Example

G.f. = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 67*x^5 + 504*x^6 + 9310*x^7 + ...

Maple

with(GroupTheory): seq(NumGroups(3^n), n=0..8); # Muniru A Asiru, Oct 17 2018

Prog

(GAP) A090091 := List([0..7], n -> NumberSmallGroups(3^n)); # Muniru A Asiru, Oct 15 2017

Crossrefs

Cf. A000001, A000679, A090130, A090140.

Sequence in context: A268116 A268123 A158079 * A268122 A133645 A090130

Adjacent sequences: A090088 A090089 A090090 * A090092 A090093 A090094

Keyword

nonn,hard,more

Author

Eamonn O'Brien (obrien(AT)math.auckland.ac.nz), Jan 22 2004

Extensions

a(7) from David Radcliffe, Feb 24 2010

a(8) from Muniru A Asiru, Oct 17 2018

a(9) from David Burrell, Sep 01 2023

Status

approved