|
|
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.
|
|
0
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
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.
|
|
LINKS
|
|
|
EXAMPLE
|
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.
|
|
PROG
|
(GAP4)
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") ;
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|