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.

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

Author

Benoit Jubin, May 12 2008

Keywords

• easy
• nonn
• tabf

Comments

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.

Examples

			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.
		

Links

• 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.

Programs

GAP

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