

A000001


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


159



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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[n1].  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., SpringerVerlag, 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, AddisonWesley Publ. Co., Reading, MA, 1989, Section 6.6 'Fibonacci Numbers' pp. 281283.
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. 169172 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), 437442, de Gruyter, BerlinNew 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

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]


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(p1,3) + gcd(p1,4), 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,q1) = 1, 2 if gcd(p,q1) = 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
 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

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.


KEYWORD

nonn,core,nice,hard,changed


AUTHOR



EXTENSIONS



STATUS

approved



