# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a000001 Showing 1-1 of 1 %I A000001 M0098 N0035 #289 Apr 28 2024 09:51:07 %S A000001 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 A000001 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 A000001 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 A000001 Number of groups of order n. %C A000001 Also, number of nonisomorphic subgroups of order n in symmetric group S_n. - _Lekraj Beedassy_, Dec 16 2004 %C A000001 Also, number of nonisomorphic primitives of the combinatorial species Lin[n-1]. - _Nicolae Boicu_, Apr 29 2011 %C A000001 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 A000001 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 A000001 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 A000001 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 A000001 S. R. Blackburn, P. M. Neumann, and G. Venkataraman, Enumeration of Finite Groups, Cambridge, 2007. %D A000001 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 302, #35. %D A000001 J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 209. %D A000001 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 A000001 CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 150. %D A000001 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 A000001 M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964. %D A000001 D. Joyner, 'Adventures in Group Theory', Johns Hopkins Press. Pp. 169-172 has table of groups of orders < 26. %D A000001 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.24, p. 481. %D A000001 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 A000001 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A000001 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A000001 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 A000001 H. A. Bender, A determination of the groups of order p^5, Ann. of Math. (2) 29, pp. 61-72 (1927). %H A000001 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 A000001 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 A000001 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 A000001 H. U. Besche, B. Eick, E. A. O'Brien and Max Horn, The Small Groups Library %H A000001 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 A000001 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. %H A000001 Henry Bottomley, Illustration of initial terms %H A000001 David Burrell, On the number of groups of order 1024, Communications in Algebra, 2021, 1-3. %H A000001 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. %H A000001 Yang-Hui He and Minhyong Kim, Learning Algebraic Structures: Preliminary Investigations, arXiv:1905.02263 [cs.LG], 2019. %H A000001 Otto Hölder, Die Gruppen der Ordnungen p^3, pq^2, pqr, p^4, Math. Ann. 43 pp. 301-412 (1893). %H A000001 Max Horn, Numbers of isomorphism types of finite groups of given order %H A000001 Rodney James, The groups of order p^6 (p an odd prime), Math. Comp. 34 (1980), 613-637. %H A000001 Rodney James and John Cannon, Computation of isomorphism classes of p-groups, Mathematics of Computation 23.105 (1969): 135-140. %H A000001 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). %H A000001 Desmond MacHale, Are There More Finite Rings than Finite Groups?, Amer. Math. Monthly (2020) Vol. 127, Issue 10, 936-938. %H A000001 Mehdi Makhul, Josef Schicho, and Audie Warren, On Galois groups of type-1 minimally rigid graphs, arXiv:2306.04392 [math.CO], 2023. %H A000001 G. A. Miller, Determination of all the groups of order 64, Amer. J. Math., 52 (1930), 617-634. %H A000001 László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2. %H A000001 Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003. %H A000001 Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)] %H A000001 D. S. Rajan, The equations D^kY=X^n in combinatorial species, Discrete Mathematics 118 (1993) 197-206 North-Holland. %H A000001 E. Rodemich, The groups of order 128, J. Algebra 67 (1980), no. 1, 129-142. %H A000001 Gordon Royle, Combinatorial Catalogues. See subpage "Generators of small groups" for explicit generators for most groups of even order < 1000. [broken link] %H A000001 D. Rusin, Asymptotics [Cached copy of lost web page] %H A000001 Eric Weisstein's World of Mathematics, Finite Group %H A000001 Wikipedia, Finite group %H A000001 M. Wild, The groups of order sixteen made easy, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31. %H A000001 Gang Xiao, SmallGroup %H A000001 Index entries for sequences related to groups %H A000001 Index entries for "core" sequences %F A000001 From _Mitch Harris_, Oct 25 2006: (Start) %F A000001 For p, q, r primes: %F A000001 a(p) = 1, a(p^2) = 2, a(p^3) = 5, a(p^4) = 14, if p = 2, otherwise 15. %F A000001 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 A000001 a(p^e) ~ p^((2/27)e^3 + O(e^(8/3))). %F A000001 a(p*q) = 1 if gcd(p,q-1) = 1, 2 if gcd(p,q-1) = p. (p < q) %F A000001 a(p*q^2) is one of the following: %F A000001 --------------------------------------------------------------------------- %F A000001 | a(p*q^2) | p*q^2 of the form | Sequences (p*q^2) | %F A000001 ---------- ------------------------------------------ --------------------- %F A000001 | (p+9)/2 | q == 1 (mod p), p odd | A350638 | %F A000001 | 5 | p=3, q=2 => p*q^2 = 12 |Special case with A_4| %F A000001 | 5 | p=2, q odd | A143928 | %F A000001 | 5 | p == 1 (mod q^2) | A350115 | %F A000001 | 4 | p == 1 (mod q), p > 3, p !== 1 (mod q^2) | A349495 | %F A000001 | 3 | q == -1 (mod p), p and q odd | A350245 | %F A000001 | 2 | q !== +-1 (mod p) and p !== 1 (mod q) | A350422 | %F A000001 --------------------------------------------------------------------------- %F A000001 [Table from _Bernard Schott_, Jan 18 2022] %F A000001 a(p*q*r) (p < q < r) is one of the following: %F A000001 q == 1 (mod p) r == 1 (mod p) r == 1 (mod q) a(p*q*r) %F A000001 -------------- -------------- -------------- -------- %F A000001 No No No 1 %F A000001 No No Yes 2 %F A000001 No Yes No 2 %F A000001 No Yes Yes 4 %F A000001 Yes No No 2 %F A000001 Yes No Yes 3 %F A000001 Yes Yes No p+2 %F A000001 Yes Yes Yes p+4 %F A000001 [table from Derek Holt]. %F A000001 (End) %F A000001 a(n) = A000688(n) + A060689(n). - _R. J. Mathar_, Mar 14 2015 %e A000001 Groups of orders 1 through 10 (C_n = cyclic, D_n = dihedral of order n, Q_8 = quaternion, S_n = symmetric): %e A000001 1: C_1 %e A000001 2: C_2 %e A000001 3: C_3 %e A000001 4: C_4, C_2 X C_2 %e A000001 5: C_5 %e A000001 6: C_6, S_3=D_6 %e A000001 7: C_7 %e A000001 8: C_8, C_4 X C_2, C_2 X C_2 X C_2, D_8, Q_8 %e A000001 9: C_9, C_3 X C_3 %e A000001 10: C_10, D_10 %p A000001 GroupTheory:-NumGroups(n); # with(GroupTheory); loads this command - _N. J. A. Sloane_, Dec 28 2017 %t A000001 FiniteGroupCount[Range[100]] (* _Harvey P. Dale_, Jan 29 2013 *) %t A000001 a[ n_] := If[ n < 1, 0, FiniteGroupCount @ n]; (* _Michael Somos_, May 28 2014 *) %o A000001 (Magma) D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n) : n in [1..1000] ]; // _John Cannon_, Dec 23 2006 %o A000001 (GAP) A000001 := Concatenation([0], List([1..500], n -> NumberSmallGroups(n))); # _Muniru A Asiru_, Oct 15 2017 %Y A000001 The main sequences concerned with group theory are A000001 (this one), A000679, A001034, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A000688, A060689, A051532. %Y A000001 Cf. A046058, A046059, A023675, A023676. %Y A000001 A003277 gives n for which A000001(n) = 1, A063756 (partial sums). %Y A000001 A046057 gives first occurrence of each k. %Y A000001 A027623 gives the number of rings of order n. %K A000001 nonn,core,nice,hard %O A000001 0,5 %A A000001 _N. J. A. Sloane_ %E A000001 More terms from _Michael Somos_ %E A000001 Typo in b-file description fixed by _David Applegate_, Sep 05 2009 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE