%I M0098 N0035 #289 Apr 28 2024 09:51:07

%S 0,1,1,1,2,1,2,1,5,2,2,1,5,1,2,1,14,1,5,1,5,2,2,1,15,2,2,5,4,1,4,1,51,

%T 1,2,1,14,1,2,2,14,1,6,1,4,2,2,1,52,2,5,1,5,1,15,2,13,2,2,1,13,1,2,4,

%U 267,1,4,1,5,1,4,1,50,1,2,3,4,1,6,1,52,15,2,1,15,1,2,1,12,1,10,1,4,2

%N Number of groups of order n.

%C Also, number of nonisomorphic subgroups of order n in symmetric group S_n. - _Lekraj Beedassy_, Dec 16 2004

%C Also, number of nonisomorphic primitives of the combinatorial species Lin[n-1]. - _Nicolae Boicu_, Apr 29 2011

%C In (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008), a(n) is called the "group number of n", denoted by gnu(n), and the first occurrence of k is called the "minimal order attaining k", denoted by moa(k) (see A046057). - _Daniel Forgues_, Feb 15 2017

%C It is conjectured in (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008) that the sequence n -> a(n) -> a(a(n)) = a^2(n) -> a(a(a(n))) = a^3(n) -> ... -> consists ultimately of 1s, where a(n), denoted by gnu(n), is called the "group number of n". - _Muniru A Asiru_, Nov 19 2017

%C MacHale (2020) shows that there are infinitely many values of n for which there are more groups than rings of that order (cf. A027623). He gives n = 36355 as an example. It would be nice to have enough values of n to create an OEIS entry for them. - _N. J. A. Sloane_, Jan 02 2021

%C I conjecture that a(i) * a(j) <= a(i*j) for all nonnegative integers i and j. - _Jorge R. F. F. Lopes_, Apr 21 2024

%D S. R. Blackburn, P. M. Neumann, and G. Venkataraman, Enumeration of Finite Groups, Cambridge, 2007.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 302, #35.

%D J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 209.

%D H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.

%D CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 150.

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, A Foundation for Computer Science, Addison-Wesley Publ. Co., Reading, MA, 1989, Section 6.6 'Fibonacci Numbers' pp. 281-283.

%D M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964.

%D D. Joyner, 'Adventures in Group Theory', Johns Hopkins Press. Pp. 169-172 has table of groups of orders < 26.

%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.24, p. 481.

%D M. F. Newman and E. A. O'Brien, A CAYLEY library for the groups of order dividing 128. Group theory (Singapore, 1987), 437-442, de Gruyter, Berlin-New York, 1989.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H H.-U. Besche and Ivan Panchenko, <a href="/A000001/b000001.txt">Table of n, a(n) for n = 0..2047</a> [Terms 1 through 2015 copied from Small Groups Library mentioned below. Terms 2016 - 2047 added by _Ivan Panchenko_, Aug 29 2009. 0 prepended by _Ray Chandler_, Sep 16 2015. a(1024) corrected by _Benjamin Przybocki_, Jan 06 2022]

%H H. A. Bender, <a href="http://www.jstor.org/stable/1967981">A determination of the groups of order p^5</a>, Ann. of Math. (2) 29, pp. 61-72 (1927).

%H Hans Ulrich Besche and Bettina Eick, <a href="http://dx.doi.org/10.1006/jsco.1998.0258">Construction of finite groups</a>, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.

%H Hans Ulrich Besche and Bettina Eick, <a href="http://dx.doi.org/10.1006/jsco.1998.0259">The groups of order at most 1000 except 512 and 768</a>, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 405-413.

%H H. U. Besche, B. Eick and E. A. O'Brien, <a href="http://www.ams.org/era/2001-07-01/S1079-6762-01-00087-7/home.html">The groups of order at most 2000</a>, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1-4.

%H H. U. Besche, B. Eick, E. A. O'Brien and Max Horn, <a href="https://gap-packages.github.io/smallgrp/">The Small Groups Library</a>

%H H. U. Besche, B. Eick and E. A. O'Brien, <a href="https://web.archive.org/web/20161030100727/http://www.icm.tu-bs.de/ag_algebra/software/small/number.html">Number of isomorphism types of finite groups of given order</a> [gives incorrect a(1024)]

%H H.-U. Besche, B. Eick and E. A. O'Brien, <a href="http://dx.doi.org/10.1142/S0218196702001115">A Millennium Project: Constructing Small Groups</a>, Internat. J. Algebra and Computation, 12 (2002), 623-644.

%H Henry Bottomley, <a href="/A000001/a000001.gif">Illustration of initial terms</a>

%H David Burrell, <a href="https://doi.org/10.1080/00927872.2021.2006680">On the number of groups of order 1024</a>, Communications in Algebra, 2021, 1-3.

%H J. H. Conway, Heiko Dietrich and E. A. O'Brien, <a href="http://www.math.auckland.ac.nz/~obrien/research/gnu.pdf">Counting groups: gnus, moas and other exotica</a>, Math. Intell., Vol. 30, No. 2, Spring 2008.

%H Yang-Hui He and Minhyong Kim, <a href="https://arxiv.org/abs/1905.02263">Learning Algebraic Structures: Preliminary Investigations</a>, arXiv:1905.02263 [cs.LG], 2019.

%H Otto Hölder, <a href="http://dx.doi.org/10.1007/BF01443651">Die Gruppen der Ordnungen p^3, pq^2, pqr, p^4</a>, Math. Ann. 43 pp. 301-412 (1893).

%H Max Horn, <a href="https://groups.quendi.de/">Numbers of isomorphism types of finite groups of given order</a>

%H Rodney James, <a href="http://dx.doi.org/10.1090/S0025-5718-1980-0559207-0">The groups of order p^6 (p an odd prime)</a>, Math. Comp. 34 (1980), 613-637.

%H Rodney James and John Cannon, <a href="http://dx.doi.org/10.1090/S0025-5718-1969-0238953-8">Computation of isomorphism classes of p-groups</a>, Mathematics of Computation 23.105 (1969): 135-140.

%H Olexandr Konovalov, <a href="https://github.com/olexandr-konovalov/gnu/tree/master">Crowdsourcing project for the database of numbers of isomorphism types of finite groups</a>, Github (a list of gnu(n) for many n < 50000).

%H Desmond MacHale, <a href="https://doi.org/10.1080/00029890.2020.1820790">Are There More Finite Rings than Finite Groups?</a>, Amer. Math. Monthly (2020) Vol. 127, Issue 10, 936-938.

%H Mehdi Makhul, Josef Schicho, and Audie Warren, <a href="https://arxiv.org/abs/2306.04392">On Galois groups of type-1 minimally rigid graphs</a>, arXiv:2306.04392 [math.CO], 2023.

%H G. A. Miller, <a href="http://www.jstor.org/stable/2370630">Determination of all the groups of order 64</a>, Amer. J. Math., 52 (1930), 617-634.

%H László Németh and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Nemeth/nemeth8.html">Sequences Involving Square Zig-Zag Shapes</a>, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.

%H Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/MAA/07-Sequence%20Pictures/mathgames_12_08_03.html">Sequence Pictures</a>, Math Games column, Dec 08 2003.

%H Ed Pegg, Jr., <a href="/A000043/a000043_2.pdf">Sequence Pictures</a>, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]

%H D. S. Rajan, <a href="http://dx.doi.org/10.1016/0012-365X(93)90061-W">The equations D^kY=X^n in combinatorial species</a>, Discrete Mathematics 118 (1993) 197-206 North-Holland.

%H E. Rodemich, <a href="http://dx.doi.org/10.1016/0021-8693(90)90244-I">The groups of order 128</a>, J. Algebra 67 (1980), no. 1, 129-142.

%H Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/data.html">Combinatorial Catalogues</a>. See subpage "Generators of small groups" for explicit generators for most groups of even order < 1000. [broken link]

%H D. Rusin, <a href="/A000001/a000001.txt">Asymptotics</a> [Cached copy of lost web page]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FiniteGroup.html">Finite Group</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Finite_group">Finite group</a>

%H M. Wild, <a href="http://www.jstor.org/stable/30037381">The groups of order sixteen made easy</a>, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.

%H Gang Xiao, <a href="http://wims.unice.fr/~wims/wims.cgi?module=tool/algebra/smallgroup">SmallGroup</a>

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%F From _Mitch Harris_, Oct 25 2006: (Start)

%F For p, q, r primes:

%F a(p) = 1, a(p^2) = 2, a(p^3) = 5, a(p^4) = 14, if p = 2, otherwise 15.

%F a(p^5) = 61 + 2*p + 2*gcd(p-1,3) + gcd(p-1,4), p >= 5, a(2^5)=51, a(3^5)=67.

%F a(p^e) ~ p^((2/27)e^3 + O(e^(8/3))).

%F a(p*q) = 1 if gcd(p,q-1) = 1, 2 if gcd(p,q-1) = p. (p < q)

%F a(p*q^2) is one of the following:

%F ---------------------------------------------------------------------------

%F | a(p*q^2) | p*q^2 of the form | Sequences (p*q^2) |

%F ---------- ------------------------------------------ ---------------------

%F | (p+9)/2 | q == 1 (mod p), p odd | A350638 |

%F | 5 | p=3, q=2 => p*q^2 = 12 |Special case with A_4|

%F | 5 | p=2, q odd | A143928 |

%F | 5 | p == 1 (mod q^2) | A350115 |

%F | 4 | p == 1 (mod q), p > 3, p !== 1 (mod q^2) | A349495 |

%F | 3 | q == -1 (mod p), p and q odd | A350245 |

%F | 2 | q !== +-1 (mod p) and p !== 1 (mod q) | A350422 |

%F ---------------------------------------------------------------------------

%F [Table from _Bernard Schott_, Jan 18 2022]

%F a(p*q*r) (p < q < r) is one of the following:

%F q == 1 (mod p) r == 1 (mod p) r == 1 (mod q) a(p*q*r)

%F -------------- -------------- -------------- --------

%F No No No 1

%F No No Yes 2

%F No Yes No 2

%F No Yes Yes 4

%F Yes No No 2

%F Yes No Yes 3

%F Yes Yes No p+2

%F Yes Yes Yes p+4

%F [table from Derek Holt].

%F (End)

%F a(n) = A000688(n) + A060689(n). - _R. J. Mathar_, Mar 14 2015

%e Groups of orders 1 through 10 (C_n = cyclic, D_n = dihedral of order n, Q_8 = quaternion, S_n = symmetric):

%e 1: C_1

%e 2: C_2

%e 3: C_3

%e 4: C_4, C_2 X C_2

%e 5: C_5

%e 6: C_6, S_3=D_6

%e 7: C_7

%e 8: C_8, C_4 X C_2, C_2 X C_2 X C_2, D_8, Q_8

%e 9: C_9, C_3 X C_3

%e 10: C_10, D_10

%p GroupTheory:-NumGroups(n); # with(GroupTheory); loads this command - _N. J. A. Sloane_, Dec 28 2017

%t FiniteGroupCount[Range[100]] (* _Harvey P. Dale_, Jan 29 2013 *)

%t a[ n_] := If[ n < 1, 0, FiniteGroupCount @ n]; (* _Michael Somos_, May 28 2014 *)

%o (Magma) D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n) : n in [1..1000] ]; // _John Cannon_, Dec 23 2006

%o (GAP) A000001 := Concatenation([0], List([1..500], n -> NumberSmallGroups(n))); # _Muniru A Asiru_, Oct 15 2017

%Y The main sequences concerned with group theory are A000001 (this one), A000679, A001034, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A000688, A060689, A051532.

%Y Cf. A046058, A046059, A023675, A023676.

%Y A003277 gives n for which A000001(n) = 1, A063756 (partial sums).

%Y A046057 gives first occurrence of each k.

%Y A027623 gives the number of rings of order n.

%K nonn,core,nice,hard

%O 0,5

%A _N. J. A. Sloane_

%E More terms from _Michael Somos_

%E Typo in b-file description fixed by _David Applegate_, Sep 05 2009