A001034 -id:A001034

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A000001

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

+10
162

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[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

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*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=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.

KEYWORD

nonn,core,nice,hard

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

A005180

Orders of simple groups.
(Formerly M0651)

+10
34

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 60, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 168, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

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

OFFSET

1,2

COMMENTS

Officially the group of order 1 is not considered to be simple - see for example Rotman, Group Theory.

REFERENCES

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].

M. Hall, Jr., A search for simple groups of order less than one million, pp. 137-168 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.

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

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

M. Aschbacher, Review of "Classification of the finite simple groups, No. 2" by D. Gorenstein, R. Lyons & R. Solomon

C. Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574-577.

Dept. of Pure Math., Univ. Sheffield, The Classification of Finite Simple Groups

D. Gorenstein, R. Lyons & R. Solomon, The Classification of the Finite Simple Groups, AMS Books Online, Providence RI 1994.

D. Gorenstein, R. Lyons & R. Solomon, The Classification of the Finite Simple Groups, Number 2, AMS Books Online, Providence RI 1996.

R. K. Guy and N. J. A. Sloane, Correspondence, 1988.

R. Solomon, A Brief History Of The Classification Of The Finite Simple Groups, Bull. Amer. Math. Soc. 38 (2001), 315-352.

Index entries for sequences related to groups

MATHEMATICA

(* Recomputation from A001034. *)

maxOrder = 7789;

A001034 = Select[Cases[Import["https://oeis.org/A001034/b001034.txt", "Table"], {_, _}][[All, 2]], # <= maxOrder&];

Union[{1}, Prime[Range[PrimePi[maxOrder]]], A001034] (* Jean-François Alcover, Aug 19 2019 *)

CROSSREFS

Cf. A000001, A001228. Union of {1}, A000040 and A001034.

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane, R. K. Guy

STATUS

approved

A001228

Orders of sporadic simple groups.

+10
24

7920, 95040, 175560, 443520, 604800, 10200960, 44352000, 50232960, 244823040, 898128000, 4030387200, 145926144000, 448345497600, 460815505920, 495766656000, 42305421312000, 64561751654400, 273030912000000, 51765179004000000, 90745943887872000, 4089470473293004800, 4157776806543360000, 86775571046077562880, 1255205709190661721292800, 4154781481226426191177580544000000, 808017424794512875886459904961710757005754368000000000

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

OFFSET

1,1

COMMENTS

Numbers of divisors: A174601(n) = A000005(a(n)); squarefree kernels: A174848(n) = A007947(a(n)). - Reinhard Zumkeller, Apr 02 2010

By historical convention, the Tits group is often excluded from the list of sporadic simple groups. It could be inserted as a(7) = 17971200 giving this sequence 27 rather than 26 elements. - Charles R Greathouse IV, Jul 09 2020

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 296.

Martin Gardner, "The Last Recreations", 1997, chap 9, p. 153.

LINKS

Table of n, a(n) for n=1..26.

David Madore, Table of sporadic simple groups.

Grant Sanderson, Group theory and 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000, 3Blue1Brown video (2020)

Eric Weisstein's World of Mathematics, Sporadic Group

Index entries for sequences related to groups

EXAMPLE

The first term is 7920 because the order of the sporadic group M_{11} is 7920, the smallest order of any sporadic group.

CROSSREFS

Cf. A001034, A005180.

KEYWORD

nonn,fini,nice,full

AUTHOR

N. J. A. Sloane

EXTENSIONS

Entries checked by Pab Ter (pabrlos(AT)yahoo.com), May 29 2004

STATUS

approved

A056866

Orders of non-solvable groups, i.e., numbers that are not solvable numbers.

+10
18

60, 120, 168, 180, 240, 300, 336, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 900, 960, 1008, 1020, 1080, 1092, 1140, 1176, 1200, 1260, 1320, 1344, 1380, 1440, 1500, 1512, 1560, 1620, 1680, 1740, 1800, 1848, 1860, 1920, 1980, 2016, 2040

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

OFFSET

1,1

COMMENTS

A number is solvable if every group of that order is solvable.

This comment is about the three sequences A001034, A060793, A056866: The Feit-Thompson theorem says that a finite group with odd order is solvable, hence all numbers in this sequence are even. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001 [Corrected by Isaac Saffold, Aug 09 2021]

Insoluble group orders can be derived from A001034 (simple non-cyclic orders): k is an insoluble order iff k is a multiple of a simple non-cyclic order. - Des MacHale

All terms are divisible by 4 and either 3 or 5. - Charles R Greathouse IV, Sep 11 2012

Subsequence of A056868 and hence of A060652. - Charles R Greathouse IV, Apr 16 2015, updated Sep 11 2015

The primitive elements are A257146. Since the sum of the reciprocals of the terms of that sequence converges, this sequence has a natural density and so a(n) ~ k*n for some k (see, e.g., Erdős 1948). - Charles R Greathouse IV, Apr 17 2015

From Jianing Song, Apr 04 2022: (Start)

Burnside's p^a*q^b theorem says that a finite group whose order has at most 2 distinct prime factors is solvable, hence all terms have at least 3 distinct prime factors.

Terms not divisible by 12 are divisible by 320 and have at least 4 distinct prime factors (cf. A257391). (End)

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2240 terms from T. D. Noe)

R. Brauer, Investigation on groups of even order, I.

R. Brauer, Investigation on groups of even order, II.

P. Erdős, On the density of some sequences of integers, Bull. Amer. Math. Soc. 54 (1948), pp. 685-692. See p. 685.

W. Feit and J. G. Thompson, A solvability criterion for finite groups and consequences, Proc. N. A. S. 48 (6) (1962) 968.

J. Pakianathan and K. Shankar, Nilpotent numbers, Amer. Math. Monthly, 107, August-September 2000, 631-634.

Cindy Tsang, Qin Chao, On the solvability of regular subgroups in the holomorph of a finite solvable group, arXiv:1901.10636 [math.GR], 2019.

Index entries for sequences related to groups

FORMULA

A positive integer k is a non-solvable number if and only if it is a multiple of any of the following numbers: a) 2^p*(2^(2*p)-1), p any prime. b) 3^p*(3^(2*p)-1)/2, p odd prime. c) p*(p^2-1)/2, p prime greater than 3 such that p^2 + 1 == 0 (mod 5). d) 2^4*3^3*13. e) 2^(2*p)*(2^(2*p)+1)*(2^p-1), p odd prime.

MATHEMATICA

ma[n_] := For[k = 1, True, k++, p = Prime[k]; m = 2^p*(2^(2*p) - 1); If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]; mb[n_] := For[k = 2, True, k++, p = Prime[k]; m = 3^p*((3^(2*p) - 1)/2); If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]; mc[n_] := For[k = 3, True, k++, p = Prime[k]; m = p*((p^2 - 1)/2); If[Mod[p^2 + 1, 5] == 0, If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]]; md[n_] := Mod[n, 2^4*3^3*13] == 0; me[n_] := For[k = 2, True, k++, p = Prime[k]; m = 2^(2*p)*(2^(2*p) + 1)*(2^p - 1); If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]; notSolvableQ[n_] := OddQ[n] || ma[n] || mb[n] || mc[n] || md[n] || me[n]; Select[ Range[3000], notSolvableQ] (* Jean-François Alcover, Jun 14 2012, from formula *)

PROG

(PARI) is(n)={

if(n%5616==0, return(1));

forprime(p=2, valuation(n, 2),

if(n%(4^p-1)==0, return(1))

);

forprime(p=3, valuation(n, 3),

if(n%(9^p\2)==0, return(1))

);

forprime(p=3, valuation(n, 2)\2,

if(n%((4^p+1)*(2^p-1))==0, return(1))

);

my(f=factor(n)[, 1]);

for(i=1, #f,

if(f[i]>3 && f[i]%5>1 && f[i]%5<4 && n%(f[i]^2\2)==0, return(1))

);

}; \\ Charles R Greathouse IV, Sep 11 2012

CROSSREFS

Subsequence of A000977 and A056868.

Cf. A003277, A051532, A056867, A001034, A060652.

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Sep 02 2000

EXTENSIONS

More terms from Des MacHale, Feb 19 2001

Further terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

STATUS

approved

A060793

Orders of finite perfect groups (groups such that G = G' where G' is the commutator subgroup of G).

+10
13

1, 60, 120, 168, 336, 360, 504, 660, 720, 960, 1080, 1092, 1320, 1344, 1920, 2160, 2184, 2448, 2520, 2688, 3000, 3420, 3600, 3840, 4080, 4860, 4896, 5040, 5376, 5616, 5760, 6048, 6072, 6840, 7200, 7500, 7560, 7680, 7800, 7920, 9720, 9828, 10080, 10752

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

OFFSET

1,2

COMMENTS

Since a non-cyclic simple group is perfect this sequence contains A001034 and since a perfect group is non-solvable this sequence is a subsequence of A056866 (apart from the initial term).

REFERENCES

D. Holt and W. Plesken, Perfect Groups, Oxford University Press, 1989.

LINKS

Eric M. Schmidt, Table of n, a(n) for n = 1..300

Walter Feit, J. G. Thompson, A solvability criterion for finite groups and some consequences, Proc. N. A. S. 48 (6) (1962) 968.

Index entries for sequences related to groups

EXAMPLE

A_{5} is perfect since it is equivalent to A_{5}'.

PROG

(GAP) SizesPerfectGroups(); # Eric M. Schmidt, Nov 14 2013

CROSSREFS

Cf. A001034, A056866.

KEYWORD

nonn

AUTHOR

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 26 2001

STATUS

approved

A109379

Orders of non-cyclic simple groups (with repetition).

+10
7

60, 168, 360, 504, 660, 1092, 2448, 2520, 3420, 4080, 5616, 6048, 6072, 7800, 7920, 9828, 12180, 14880, 20160, 20160, 25308, 25920, 29120, 32736, 34440, 39732, 51888, 58800, 62400, 74412, 95040, 102660, 113460, 126000, 150348

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

OFFSET

1,1

COMMENTS

The first repetition is at 20160 (= 8!/2) and the first proof that there exist two nonisomorphic simple groups of this order was given by the American mathematician Ida May Schottenfels (1869-1942). - David Callan, Nov 21 2006

By the Feit-Thompson theorem, all terms in this sequence are even. - Robin Jones, Dec 25 2023

REFERENCES

See A001034 for references and other links.

LINKS

David A. Madore, Table of n, a(n) for n = 1..493 [taken from link below]

David A. Madore, More terms

John McKay, The non-abelian simple groups g, |g|<10^6 - character tables, Commun. Algebra 7 (1979) no. 13, 1407-1445.

Ida May Schottenfels, Two Non-Isomorphic Simple Groups of the Same Order 20,160, Annals of Math., 2nd Ser., Vol. 1, No. 1/4 (1899), pp. 147-152.

Wikipedia, Feit-Thompson theorem.

Index entries for sequences related to groups

CROSSREFS

Cf. A000001, A000679, A005180, A001228, A060793, A056866, A056868, A119630.

Cf. A001034 (orders without repetition), A119648 (orders that are repeated).

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, Jul 29 2006

STATUS

approved

A119630

Orders of non-Abelian simple groups of rank at least two.

+10
5

60, 168, 360, 2520, 5616, 6048, 7920, 20160, 20160, 25920, 29120, 62400, 95040, 126000, 175560, 181440, 372000, 443520, 604800, 979200, 1451520, 1814400, 1876896, 3265920, 4245696, 4680000, 5515776, 5663616, 6065280, 9999360

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

OFFSET

1,1

COMMENTS

This omits (from A001034) only the non-Abelian simple groups of Lie type of rank 1, i.e. the linear groups L(2,q).

LINKS

David A. Madore, Table of n, a(n) for n = 1..243 [taken from link below]

David A. Madore, More terms

Index entries for sequences related to groups

CROSSREFS

Subsequence of A005180.

Cf. A001034.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jun 10 2006

STATUS

approved

A131932

Number of nonisomorphic nonsolvable groups of order A056866(n).

+10
4

1, 3, 1, 1, 8, 1, 3, 6, 1, 26, 2, 2, 5, 2, 8, 23, 1, 6, 1, 107, 6, 1, 14, 1, 1, 1, 19, 2, 8, 28, 1, 93, 2, 4, 5, 5, 22, 1, 10, 1, 1, 588, 2, 20, 5, 1, 64, 4, 1, 5, 2, 5, 81, 1, 1, 18, 1, 25, 112, 2, 5, 1, 1

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..63.

O. L. Hoelder, Bildung zusammengesetzter Gruppen, Math. Ann., 46 (1895), 321-422; see p. 420.

EXAMPLE

a(1) = 1 because there is only 1 nonsolvable group of order 60: A_5 (alternating group of 5th degree).

a(2) = 3 because there are 3 different nonsolvable groups of order 120.

PROG

(GAP) NrUnsolvable := function(n) local i, count; count := 0; for i in [1..NumberSmallGroups(n)] do if not IsSolvableGroup(SmallGroup(n, i)) then count := count + 1; fi; od; return count; end; # Eric M. Schmidt, Apr 04 2013

(GAP) LoadPackage("GrpConst"); NrUnsolvable := function(n) local i, j, num; num := 0; for i in DivisorsInt(n) do if i<>1 then for j in [1..NrPerfectGroups(i)] do num := num + Length(Remove(UpwardsExtensions(PerfectGroup(IsPermGroup, i, j), n/i))); od; fi; od; return num; end; # Eric M. Schmidt, Nov 14 2013

CROSSREFS

Cf. A001034, A060793, A056866, A056868.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Jul 30 2007, Oct 20 2007

EXTENSIONS

Edited by N. J. A. Sloane, Oct 08 2007

More terms from Eric M. Schmidt, Apr 04 2013

a(44)-a(63) from Eric M. Schmidt, Nov 14 2013

STATUS

approved

A001051

Number of subgroups of order n in orthogonal group O(3).

+10
2

1, 3, 1, 5, 1, 5, 1, 7, 1, 5, 1, 8, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 10, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 8, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 8, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 8

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

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Orthogonal Group.

FORMULA

Has period 1 5 1 7 except that a(2) = 3, a(4) = 5, a(12) = 8, a(24) = 10, a(48) = a(60) = a(120) = 8.

MATHEMATICA

a[2] = 3; a[4] = 5; a[12] = 8; a[24] = 10; a[48] = a[60] = a[120] = 8; a[n_] := Switch[Mod[n, 4], 0, 7, 1, 1, 2, 5, 3, 1]; Table[a[n], {n, 1, 96}] (* Jean-François Alcover, Oct 15 2013 *)

PROG

(PARI) A001051(n) = if((12==n)||(48==n)||(60==n)||(120==n), 8, if(24==n, 10, if((4==n)||(2==n), 1+n, [1, 5, 1, 7][1+((n-1)%4)]))); \\ Antti Karttunen, Jan 15 2019

CROSSREFS

The main sequences concerned with group theory are A000001, A000679, A001034, A001051, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A051881.

KEYWORD

nonn,easy,nice

AUTHOR

J. H. Conway

EXTENSIONS

Data section extended up to a(120) by Antti Karttunen, Jan 15 2019

STATUS

approved

A051881

Number of subgroups of order n in special orthogonal group SO(3).

+10
2

1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1

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

OFFSET

1,4

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..12000

FORMULA

Has period 1, 2 except for a(2) = 1, a(12) = a(24) = a(60) = 3.

EXAMPLE

The groups are "nn", of order n; "22n", of order 2n; "332", "432", "532" of orders 12,24,60.

MATHEMATICA

a[2] = 1; a[12|24|60] = 3; a[n_] := 2-Mod[n, 2]; Array[a, 105] (* Jean-François Alcover, Nov 12 2015 *)

PROG

(PARI) a(n)=if(n==2||n==12||n==24||n==60, if(n>2, 3, 1), if(n%2, 1, 2)) \\ Charles R Greathouse IV, Nov 10 2015

(Python)

def a(n):

if n == 2:

return 1

elif n in {12, 24, 60}:

return 3

else:

return 2 - n % 2 # Paul Muljadi, Oct 21 2024

CROSSREFS

The main sequences concerned with group theory are A000001, A000679, A001034, A001051, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A051881.

KEYWORD

nonn,easy,nice

AUTHOR

J. H. Conway

EXTENSIONS

More terms from James A. Sellers and David W. Wilson, Dec 16 1999

STATUS

approved

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