A000001 - OEIS (original) (raw)

A000001

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

165

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

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

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

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

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

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.

Marek Dančo, Mikoláš Janota, Michael Codish, and João Jorge Araújo, Complete Symmetry Breaking for Finite Models, arXiv:2502.10155 [cs.LO], 2025. See p. 7. Mentions this sequence.

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

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

FORMULA

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*q^2) |

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

| (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 == 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 |

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

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)

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.

A046057 gives first occurrence of each k.

A027623 gives the number of rings of order n.

EXTENSIONS

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