A059773 -id:A059773 - cp's OEIS frontend

A034383 Number of labeled groups.

1, 2, 3, 16, 30, 480, 840, 22080, 68040, 1088640, 3991680, 259459200, 518918400, 16605388800, 163459296000, 10353459916800, 22230464256000, 1867358997504000, 6758061133824000, 648773868847104000, 5474029518397440000, 122618261212102656000

Offset: 1

Christian G. Bower

nonn

From Jianing Song, Mar 02 2024: (Start)
In other words, number of ways to define a group structure on a set of n elements. Note that for a group G, a group structure on the set G is given by mapping (x,y) to sigma^(-1)(sigma(x)*sigma(y)), where sigma is a bijection on the set G; sigma and sigma' give the same structure if and only if sigma' is the composition of a group automorphism of G and sigma.
By definition, a(n) = A034381(n) if n in A003277, otherwise a(n) > A034381(n). The indices of records of a(n)/A034381(n) among the known terms are 1, 4, 8, 16, 24, 32, 48, 64, 96, 128, 192, with a(192)/A034381(192) = 122774329/1640520 ~ 74.8.
Also by definition, a(n) >= A000001(n)*n!/A059773(n). If the conjecture A059773(2^r) = A002884(r) is true, then A059773(2^r) <= 2^(r^2), while A000001(2^r) >= 2^((2/27)*r^2*(r-6)) (see the Math Stack Exchange link below), so a(2^r)/A034381(2^r) tends to infinity quickly as r tends to infinity.
The sequence is strictly increasing for the first 256 terms (a(256) > A034381(256) > A034381(255) = a(255) since 255 is in A003277). On the other hand, assuming that A059773(2^r) = A002884(r), then a(2^20)/(2^20)! >= A000001(2^20)/A002884(20) > 99798.4, while a(2^20+1)/(2^20)! = A034381(2^20+1)/(2^20)! = (2^20+1)/phi(2^20+1) since 2^20+1 = 17*61681 is in A003277, so we would have a(2^20) > a(2^20+1). It is conjectured a(2^r) > a(2^r+1) for all sufficiently large r. (End)

Stephen A. Silver, Table of n, a(n) for n = 1..255
H. U. Besche, The Small Groups library
Math Stack Exchange, Known bounds for the number of groups of a given order.
Index entries for sequences related to groups

Cf. A000001, A034381, A034382, A058157, A058163.

A034383 := function(n) local fn, sum, k; fn := Factorial(n); sum := 0; for k in [1 .. NrSmallGroups(n)] do sum := sum + fn / Size(AutomorphismGroup(SmallGroup(n,k))); od; return sum; end; # Stephen A. Silver, Feb 10 2013

a(n) = n * A058163(n).

a(n) = Sum n!/|Aut(G)|, where the sum is taken over the different products G of cyclic groups with |G| = n.

More terms from Stephen A. Silver, Feb 10 2013

A061350 Maximal size of Aut(G) where G is a finite Abelian group of order n.

Original entry on oeis.org

1, 1, 2, 6, 4, 2, 6, 168, 48, 4, 10, 12, 12, 6, 8, 20160, 16, 48, 18, 24, 12, 10, 22, 336, 480, 12, 11232, 36, 28, 8, 30, 9999360, 20, 16, 24, 288, 36, 18, 24, 672, 40, 12, 42, 60, 192, 22, 46, 40320, 2016, 480, 32, 72, 52, 11232, 40, 1008, 36, 28, 58, 48, 60, 30, 288

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 07 2001

a(n) is multiplicative; if n = p^m is a prime power the maximal size of Aut(G) is attained by the elementary Abelian group G =(C_p)^m and then Aut(G) is GL(m,p) and a(n) = (p^m - 1)*(p^m - p)*...*(p^m - p^(m-1)). For general n the maximum will be for the direct product of the (C_p)^m over the prime powers dividing n and then the automorphism group is the direct product of the GL(m,p).

Equivalently, maximal size of Aut(G) where G is a nilpotent group of order n. - Eric M. Schmidt, Feb 27 2013

T. D. Noe, Table of n, a(n) for n=1..1024

Cf. A059773, A002884, A053290, A053292, A053293.

A061350 := proc(n) local ans, i, j; ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*(mul(ifactors(n)[2][i][1]^ifactors(n)[2][i][2] - ifactors(n)[2][i][1]^(j - 1), j = 1..ifactors(n)[2][i][2])): od: RETURN(ans) end:

a[p_?PrimeQ] := p-1; a[1] = 1; a[n_] := Times @@ (Product[#[[1]]^#[[2]] - #[[1]]^k, {k, 0, #[[2]]-1}]& /@ FactorInteger[n]); Table[a[n], {n, 1, 63}] (* Jean-François Alcover, May 21 2012, after Maple *)

More terms from Vladeta Jovovic, Jun 12 2001

A137316 Array read by rows: T(n,k) is the number of automorphisms of the k-th group of order n, where the ordering is such that the rows are nondecreasing.

Original entry on oeis.org

1, 1, 2, 2, 6, 4, 2, 6, 6, 4, 8, 8, 24, 168, 6, 48, 4, 20, 10, 4, 12, 12, 12, 24, 12, 6, 42, 8, 8, 16, 16, 16, 32, 32, 32, 32, 48, 64, 96, 192, 192, 20160, 16, 6, 12, 48, 54, 432, 18, 8, 20, 24, 40, 40, 12, 42, 10, 110, 22, 8, 16, 16, 24, 24, 24, 24, 24, 24, 48, 48, 48, 48, 144, 336

Offset: 1

Benoit Jubin, Apr 06 2008, Apr 15 2008

The length of the n-th row is A000001(n).
The largest value of the n-th row is A059773(n).
The number phi(n) = A000010(n) appears in the n-th row.

			The table begins as follows:
   1
   1
   2
   2   6
   4
   2   6
   6
   4   8   8  24 168
   6  48
   4  20
  10
   4  12  12  12  24
  12
   6  42
The first row with two numbers corresponds to the two groups of order 4, the cyclic group Z_4 and the Klein group Z_2 x Z_2, whose automorphism groups are respectively the group (Z_4)^* = Z_2 and the symmetric group S_3.

D. MacHale and R. Sheehy, Finite groups with few automorphisms, Math. Proc. Roy. Irish Acad., 104A(2) (2004), 231--238.

Cf. A064767, A060249, A060817, A062771, A060249, A002618, A061350.

# GAP 4
Print("\n") ;
for o in [ 1 .. 33 ] do
    n := NumberSmallGroups(o) ;
    og := [] ;
    for i in [1 .. n] do
        g := SmallGroup(o,i) ;
        H := AutomorphismGroup(g) ;
        ho := Order(H) ;
        Add(og,ho) ;
    od;
    Sort(og) ;
    Print(og) ;
    Print("\n") ;
od; # R. J. Mathar, Jul 13 2013

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A034383 Number of labeled groups.

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A061350 Maximal size of Aut(G) where G is a finite Abelian group of order n.

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A137316 Array read by rows: T(n,k) is the number of automorphisms of the k-th group of order n, where the ordering is such that the rows are nondecreasing.

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