A000679 -id:A000679 - cp's OEIS frontend

A252784 Number of exponent-2 class 2 groups of order 2^n.

Original entry on oeis.org

0, 0, 1, 3, 7, 22, 89, 779, 30078, 8785772, 48803495722, 1774274116992170

Offset: 0

Eric M. Schmidt, Dec 21 2014

J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, The Mathematical Intelligencer, March 2008, Volume 30, Issue 2, pp 6-15.
Bettina Eick and E. A. O'Brien, Enumerating p-groups. Group theory. J. Austral. Math. Soc. Ser. A 67 (1999), no. 2, 191-205.
E. A. O'Brien, Bibliography on the determination of finite groups.

Cf. A000679, A252785, A252786, A252787.

A001051 Number of subgroups of order n in orthogonal group O(3).

Original entry on oeis.org

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

Offset: 1

J. H. Conway

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Orthogonal Group.

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

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 *)

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

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.

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

A051881 Number of subgroups of order n in special orthogonal group SO(3).

Original entry on oeis.org

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

Offset: 1

J. H. Conway

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

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

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

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

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

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

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

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

A119648 Orders for which there is more than one simple group.

Original entry on oeis.org

20160, 4585351680, 228501000000000, 65784756654489600, 273457218604953600, 54025731402499584000, 3669292720793456064000, 122796979335906113871360, 6973279267500000000000000, 34426017123500213280276480

Offset: 1

N. J. A. Sloane, Jul 29 2006

nonn

All such orders are composite numbers (since there is only one group of any prime order).
Orders which are repeated in A109379.
Except for the first number, these are the orders of symplectic groups C_n(q)=Sp_{2n}(q), where n>2 and q is a power of an odd prime number (q=3,5,7,9,11,...). Also these are the orders of orthogonal groups B_n(q). - Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010
a(1) = 20160 = 8!/2 is the order of the alternating simple group A_8 that is isomorphic to the Lie group PSL_4(2), but 20160 is also the order of the Lie group PSL_3(4) that is not isomorphic to A_8 (see A137863). - Bernard Schott, May 18 2020

			From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010: (Start)
a(1)=|A_8|=8!/2=20160,
a(2)=|C_3(3)|=4585351680,
a(3)=|C_3(5)|=228501000000000, and
a(4)=|C_4(3)|=65784756654489600. (End)

See A001034 for references and other links.
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]. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]

C. Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574-577. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory. See also. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
W. Kimmerle et al., Composition Factors from the Group Ring and Artin's Theorem on Orders of Simple Groups, Proc. London Math. Soc., (3) 60 (1990), 89-122. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
David A. Madore, Orders of non-Abelian simple groups
Wikipedia, Classification of finite simple groups [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
Index entries for sequences related to groups

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

Cf. A001034 (orders of simple groups without repetition), A109379 (orders with repetition), A137863 (orders of simple groups which are non-cyclic and non-alternating).

sp(n, q) 1/2 q^n^2.(q^(2.i) - 1, i, 1, n) [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010] [This line contained some nonascii characters which were unreadable]

For n>1, a(n) is obtained as (1/2) q^(m^2)Prod(q^(2i)-1, i=1..m) for appropriate m>2 and q equal to a power of some odd prime number. [Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]

Extended up to the 10th term by Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010

A341824 Number of groups of order 2^n which occur as Aut(G) for some finite group G.

Original entry on oeis.org

1, 1, 2, 3, 4, 9, 14, 33

Offset: 0

Des MacHale, Feb 26 2021

The number of groups of order 2^n is A000679(n); the percentage of the 2-groups which occur as automorphism groups appears to decrease as n increases: 100, 100, 100, 60, 28.5, 17.6, 5.2, 1.4, ...

Jianing Song remarks that it is also interesting to consider infinite groups, and asks if there is an infinite group G with Aut(G) isomorphic to C_8. Des MacHale, Mar 03 2021, replies that at present this is not known. [Comment edited by N. J. A. Sloane, Mar 07 2021]

			a(5) = 9 because there are nine groups of order 32 which occur as automorphism groups of finite groups.
From _Bernard Schott_, Feb 26 2021: (Start)
Aut(C_15) = Aut(C_16) = Aut(C_20) = Aut(C_30) ~~ C_4 x C_2 where ~~ stands for "isomorphic to".
Aut(C_4 x C_2) = Aut(D_4) ~~ D_4 where D_4 is the dihedral group of the square.
Aut(C_24) ~~ C_2 x C_2 x C_2 = (C_2)^3.
There exist five groups of order 8 (A054397), the three groups C_4 x C_2, D_4, C_2 x C_2 x C_2 occur as automorphim groups of order 8, but the cyclic group C_8 and the quaternions group Q_8 never occur as Aut(G) for some finite G, so a(3) = 3. (End)

J. Flynn, D. MacHale, E. A. O'Brien and R. Sheehy, Finite Groups whose Automorphism Groups are 2-groups, Proc. Royal Irish Academy, 94A, (2) 1994, 137-145.

Cf. A000679, A054397, A340521, A341823, A341825.

a(n) <= A000679(n). - Des MacHale, Mar 02 2021

Offset modified by Jianing Song, Mar 02 2021

A094448 Number of indecomposable groups of order 2^n.

Original entry on oeis.org

0, 1, 1, 3, 8, 34, 201, 2000, 53410, 10435175, 49476809194

Offset: 0

Paul Boddington, Jun 04 2004

See A090751 for definition of indecomposable.

Cf. A000679.

Inverse Euler transform of A000679. - Franklin T. Adams-Watters, Sep 22 2006

More terms from Franklin T. Adams-Watters, Sep 22 2006

a(0) changed to 0 by Eric M. Schmidt, Jun 07 2014

A282460 Number of groups of order n^2.

Original entry on oeis.org

0, 1, 2, 2, 14, 2, 14, 2, 267, 15, 16, 2, 197, 2, 12, 6, 56092, 2, 176, 2, 221, 13, 12, 2, 8681, 15, 16, 504, 172, 2, 150, 2, 49487367289, 6, 16, 4, 3609, 2, 12, 13, 10281, 2, 228, 2, 167, 63, 12, 2, 15756130, 15, 227, 7, 219, 2, 7199, 15, 8085, 21, 16, 2, 4484

Offset: 0

Vincenzo Librandi, Feb 16 2017

nonn

David Burrell, On the number of groups of order 1024, Communications in Algebra, 2021, 1-3.
Max Horn, Numbers of isomorphism types of finite groups of given order

Cf. A000001, A000679, A140987, A141323, A235388.

D:=SmallGroupDatabase(); [0] cat [ NumberOfSmallGroups(D, n^2) : n in [1..31] ];

Join[{0}, FiniteGroupCount[Range[45]^2]]

a(n) = A000001(n^2). - R. J. Mathar, Feb 23 2017

a(32) corrected by David Burrell, Jun 07 2022

Terms a(46) and beyond from Max Horn's website added by Andrey Zabolotskiy, May 14 2023

A297420 Square of the number of groups of order n.

Original entry on oeis.org

0, 1, 1, 1, 4, 1, 4, 1, 25, 4, 4, 1, 25, 1, 4, 1, 196, 1, 25, 1, 25, 4, 4, 1, 225, 4, 4, 25, 16, 1, 16, 1, 2601, 1, 4, 1, 196, 1, 4, 4, 196, 1, 36, 1, 16, 4, 4, 1, 2704, 4, 25, 1, 25, 1, 225, 4, 169, 4, 4, 1, 169, 1, 4, 16, 71289, 1, 16, 1, 25, 1, 16, 1, 2500

Offset: 0

Vincenzo Librandi, Dec 31 2017

The record values are 1, 4, 25, 196, 225, 2601, 2704, 71289, 5419584, 3146312464, 110128506489369, 2448999521196387209521, etc. (A046058)

Vincenzo Librandi, Table of n, a(n) for n = 0..2047 [a(1024) corrected by Andrey Zabolotskiy]

Cf. A000001, A000679, A140987, A141323, A235388, A282460.

Concatenation([0], List([1..100], n -> NumberSmallGroups(n)^2)); # Muniru A Asiru, Jan 29 2018

D:=SmallGroupDatabase(); [0] cat [ NumberOfSmallGroups(D, n)^2 : n in [1..100] ];

with(GroupTheory):  0,seq(NumGroups(n)^2, n=1..100); # Muniru A Asiru, Jan 29 2018

Join[{0}, FiniteGroupCount[Range[200]]^2]

a(n) = A000001(n)^2.

Name clarified by Jon E. Schoenfield, May 24 2019

A318895 Number of isoclinism classes of the groups of order 2^n.

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 27, 115

Offset: 0

Jack W Grahl, Sep 05 2018

The concept of isoclinism was introduced in Hall (1940) and is crucial to enumerating the groups of order p^n where p is a prime.

An isoclinism exists between two groups G1 and G2 if the following holds: There is an isomorphism f between their two inner automorphism groups G1/Z(G1) and G2/Z(G2). There is an isomorphism h between their two commutator groups [G1, G1] and [G2, G2]. Lastly, f and h commute with F1 and F2, where F1 is the mapping from G1/Z(G1) x G1/Z(G1) to [G1, G1], given by a, b -> ab(a^-1)(b^-1), and F2 is defined analogously.

			There are 51 groups of order 32. These fall into 8 isoclinism classes. Therefore a(5) = 8.

P. Hall, The classification of prime-power groups, J. Reine Angew. Math. 182 (1940), 130-141.
Rodney James, M. F. Newman and E. A. O'Brien, The groups of order 128, Journal of Algebra, Volume 129, Issue 1 (1990), 136-158.
Vipul Naik, This sequence, along with other properties of groups of order 2^n

Cf. A000001, A000679. A000041 has an interpretation as the number of Abelian groups with order 2^n.

A209412 Number of nonisomorphic semigroups of order 2^n.

Original entry on oeis.org

1, 5, 188, 3684030417

Offset: 0

Jonathan Vos Post, Mar 08 2012

This is to A000679 as semigroups are to groups.

			a(3) = 3684030417 because there are 3684030417 nonisomorphic semigroups of order 2^3 = 8.

Cf. A000079, A000679 Number of groups of order 2^n, A027851 Number of nonisomorphic semigroups of order n.

a(n) = A027851(A000079(n)).

cp's OEIS Frontend

A252784 Number of exponent-2 class 2 groups of order 2^n.

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A001051 Number of subgroups of order n in orthogonal group O(3).

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A051881 Number of subgroups of order n in special orthogonal group SO(3).

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A119648 Orders for which there is more than one simple group.

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A341824 Number of groups of order 2^n which occur as Aut(G) for some finite group G.

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A094448 Number of indecomposable groups of order 2^n.

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A282460 Number of groups of order n^2.

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A297420 Square of the number of groups of order n.

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