A061350 -id:A061350 - cp's OEIS frontend

A059773 Maximum size of Aut(G) where G is a finite group of order n.

1, 1, 2, 6, 4, 6, 6, 168, 48, 20, 10, 24, 12, 42, 8, 20160, 16, 432, 18, 40, 42, 110, 22, 336, 480, 156, 11232, 84, 28, 120, 30, 9999360, 20, 272, 24, 864, 36, 342, 156, 672, 40, 252, 42, 220, 192, 506, 46, 40320, 2016, 12000, 32, 312, 52, 303264, 110, 1008

Offset: 1

Victor S. Miller, Feb 21 2001

If n = 2^k then take G to be (Z/2Z)^k, the Abelian group with n=2^k elements and characteristic two. It is generated by any k linearly independent (non-identity) elements, so the automorphism group has size (n-1)(n-2)(n-4)...(n-2^(k-1)), which grows as n^log n. I think one can show that this is optimal for n=2^k and furthermore that this has the highest rate of growth for any infinite sequence of n's. - Michael Kleber, Feb 21 2001

Equals A061350(n) for n in A056867. - Eric M. Schmidt, Mar 02 2013

			The corresponding groups are 1, Z2, Z3, (Z2)^2, Z5, S3, Z7, (Z2)^3, (Z3)^2, D5, Z11, A4, Z13, D7, Z15, (Z2)^4, Z17, ...

Eric M. Schmidt, Table of n, a(n) for n = 1..767

Cf. A061350.

A059773 := function(n) local max, f, i; if IsPrimePowerInt(n) then f := PrimePowersInt(n); return Product([0..f[2]-1], k->n-f[1]^k); fi; max := 1; for i in [1..NumberSmallGroups(n)] do max := Maximum(max, Size(AutomorphismGroup(SmallGroup(n,i)))); od; return max; end; # Eric M. Schmidt, Mar 02 2013

More terms from Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 09 2001

a(18)-a(56) from Stephen A. Silver, Feb 26 2013

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

A138565 Array read by rows: T(n,k) is the number of automorphisms of the k-th Abelian 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, 4, 8, 168, 6, 48, 4, 10, 4, 12, 12, 6, 8, 8, 16, 96, 192, 20160, 16, 6, 48, 18, 8, 24, 12, 10, 22, 8, 16, 336, 20, 480, 12, 18, 108, 11232, 12, 36, 28, 8, 30, 16, 32, 128, 384, 1536, 21504, 9999360, 20, 16, 24, 12, 36, 96, 288, 36, 18, 24, 16, 32, 672

Offset: 1

Benoit Jubin, May 12 2008

This is a subtable of A137316.
The length of the n-th row is A000688(n).
The largest value of the n-th row is A061350(n).
The number phi(n) = A000010(n) appears in the n-th row.
The number A064767(n) appears in the (n^3)-th row.
The number A062771(n) appears in the (2n)-th row.

			The table begins as follows:
1
1
2
2 6
4
2
6
4 8 168
6 48
4
10
4 12
The first row with two numbers corresponds to the two Abelian groups of order 4, the cyclic group C_4 and the Klein group C_2 x C_2, whose automorphism groups are respectively the group (C_4)^x = C_2 and the symmetric group S_3.

C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, arXiv:math/0605185 [math.GR], 2006.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, Amer. Math. Monthly 114 (2007), no 10, 917-923.
D. MacHale and R. Sheehy, Finite groups with few automorphisms, Math. Proc. Roy. Irish Acad., 104A(2) (2004), 231--238.

Print("\n") ;
for o in [ 1 .. 40 ] do
    n := NumberSmallGroups(o) ;
    og := [] ;
    for i in [1 .. n] do
        g := SmallGroup(o,i) ;
        if IsAbelian(g) then
            H := AutomorphismGroup(g) ;
            ho := Order(H) ;
            Add(og,ho) ;
        fi ;
    od;
    Sort(og) ;
    Print(og) ;
    Print("\n") ;
od; # R. J. Mathar, Jul 13 2013

cp's OEIS Frontend

A059773 Maximum size of Aut(G) where G is a finite 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

A138565 Array read by rows: T(n,k) is the number of automorphisms of the k-th Abelian group of order n, where the ordering is such that the rows are nondecreasing.

Views

Author

Keywords

Comments

Examples

Links

Programs

GAP