A000001 Number of groups of order n. (Formerly M0098 N0035)

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, 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, 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

Offset

0,5

Comments

Also, number of nonisomorphic subgroups of order n in symmetric group S_n. - Lekraj Beedassy, Dec 16 2004

Also, number of nonisomorphic primitives of the combinatorial species Lin[n-1]. - Nicolae Boicu, Apr 29 2011

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

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

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

References

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

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

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

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

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

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.

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

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

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

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.

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

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

Links

H.-U. Besche and Ivan Panchenko, Table of n, a(n) for n = 0..2047 [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. A. Bender, A determination of the groups of order p^5, Ann. of Math. (2) 29, pp. 61-72 (1927).

Hans Ulrich Besche and Bettina Eick, Construction of finite groups, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.

Hans Ulrich Besche and Bettina Eick, The groups of order at most 1000 except 512 and 768, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 405-413.

H. U. Besche, B. Eick and E. A. O'Brien, The groups of order at most 2000, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1-4.

H. U. Besche, B. Eick, E. A. O'Brien and Max Horn, The Small Groups Library

H. U. Besche, B. Eick and E. A. O'Brien, Number of isomorphism types of finite groups of given order [gives incorrect a(1024)]

H.-U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.

Henry Bottomley, Illustration of initial terms

David Burrell, On the number of groups of order 1024, Communications in Algebra, 2021, 1-3.

J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, Math. Intell., Vol. 30, No. 2, Spring 2008.

Yang-Hui He and Minhyong Kim, Learning Algebraic Structures: Preliminary Investigations, arXiv:1905.02263 [cs.LG], 2019.

Otto Hölder, Die Gruppen der Ordnungen p^3, pq^2, pqr, p^4, Math. Ann. 43 pp. 301-412 (1893).

Max Horn, Numbers of isomorphism types of finite groups of given order

Rodney James, The groups of order p^6 (p an odd prime), Math. Comp. 34 (1980), 613-637.

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

Olexandr Konovalov, Crowdsourcing project for the database of numbers of isomorphism types of finite groups, Github (a list of gnu(n) for many n < 50000).

Desmond MacHale, Are There More Finite Rings than Finite Groups?, Amer. Math. Monthly (2020) Vol. 127, Issue 10, 936-938.

Mehdi Makhul, Josef Schicho, and Audie Warren, On Galois groups of type-1 minimally rigid graphs, arXiv:2306.04392 [math.CO], 2023.

G. A. Miller, Determination of all the groups of order 64, Amer. J. Math., 52 (1930), 617-634.

László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.

Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.

Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]

D. S. Rajan, The equations D^kY=X^n in combinatorial species, Discrete Mathematics 118 (1993) 197-206 North-Holland.

E. Rodemich, The groups of order 128, J. Algebra 67 (1980), no. 1, 129-142.

Gordon Royle, Combinatorial Catalogues. See subpage "Generators of small groups" for explicit generators for most groups of even order < 1000. [broken link]

D. Rusin, Asymptotics [Cached copy of lost web page]

Eric Weisstein's World of Mathematics, Finite Group

Wikipedia, Finite group

M. Wild, The groups of order sixteen made easy, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.

Gang Xiao, SmallGroup

Index entries for sequences related to groups

Index entries for "core" sequences

Formula

From Mitch Harris, Oct 25 2006: (Start)

For p, q, r primes:

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

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.

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

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

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

---------------------------------------------------------------------------

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

| | | (p^2*q) |

---------- ------------------------------------------ ---------------------

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

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

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

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

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

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

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

---------------------------------------------------------------------------

[Table from Bernard Schott, Jan 18 2022]

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

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

-------------- -------------- -------------- --------

No No No 1

No No Yes 2

No Yes No 2

No Yes Yes 4

Yes No No 2

Yes No Yes 3

Yes Yes No p+2

Yes Yes Yes p+4

[table from Derek Holt].

(End)

a(n) = A000688(n) + A060689(n). - R. J. Mathar, Mar 14 2015

Example

Groups of orders 1 through 10 (C_n = cyclic, D_n = dihedral of order n, Q_8 = quaternion, S_n = symmetric):
1: C_1
2: C_2
3: C_3
4: C_4, C_2 X C_2
5: C_5
6: C_6, S_3=D_6
7: C_7
8: C_8, C_4 X C_2, C_2 X C_2 X C_2, D_8, Q_8
9: C_9, C_3 X C_3
10: C_10, D_10

Maple

GroupTheory:-NumGroups(n); # with(GroupTheory); loads this command - N. J. A. Sloane, Dec 28 2017

Mathematica

FiniteGroupCount[Range[100]] (* Harvey P. Dale, Jan 29 2013 *)
a[ n_] := If[ n < 1, 0, FiniteGroupCount @ n]; (* Michael Somos, May 28 2014 *)

Prog

(Magma) D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n) : n in [1..1000] ]; // John Cannon, Dec 23 2006
(GAP) A000001 := Concatenation([0], List([1..500], n -> NumberSmallGroups(n))); # Muniru A Asiru, Oct 15 2017

Crossrefs

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

Cf. A046058, A046059, A023675, A023676.

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

A046057 gives first occurrence of each k.

A027623 gives the number of rings of order n.

Sequence in context: A066083 A128644 A201733 * A172133 A146002 A109087

Adjacent sequences: * A000002 A000003 A000004 A000005 A000006 A000007

Keyword

nonn,core,nice,hard,changed

Author

N. J. A. Sloane

Extensions

More terms from Michael Somos

Typo in b-file description fixed by David Applegate, Sep 05 2009

Status

approved