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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.

%I A138565 #19 Nov 22 2024 20:06:11
%S A138565 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,
%T A138565 48,18,8,24,12,10,22,8,16,336,20,480,12,18,108,11232,12,36,28,8,30,16,
%U A138565 32,128,384,1536,21504,9999360,20,16,24,12,36,96,288,36,18,24,16,32,672
%N 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.
%C A138565 This is a subtable of A137316.
%C A138565 The length of the n-th row is A000688(n).
%C A138565 The largest value of the n-th row is A061350(n).
%C A138565 The number phi(n) = A000010(n) appears in the n-th row.
%C A138565 The number A064767(n) appears in the (n^3)-th row.
%C A138565 The number A062771(n) appears in the (2n)-th row.
%H A138565 C. J. Hillar and D. L. Rhea, <a href="http://arxiv.org/abs/math/0605185">Automorphisms of finite abelian groups</a>, arXiv:math/0605185 [math.GR], 2006.
%H A138565 C. J. Hillar and D. L. Rhea, <a href="http://www.jstor.org/stable/27642365">Automorphisms of finite abelian groups</a>, Amer. Math. Monthly 114 (2007), no 10, 917-923.
%H A138565 D. MacHale and R. Sheehy, <a href="http://www.jstor.org/stable/40656888">Finite groups with few automorphisms</a>, Math. Proc. Roy. Irish Acad., 104A(2) (2004), 231--238.
%e A138565 The table begins as follows:
%e A138565 1
%e A138565 1
%e A138565 2
%e A138565 2 6
%e A138565 4
%e A138565 2
%e A138565 6
%e A138565 4 8 168
%e A138565 6 48
%e A138565 4
%e A138565 10
%e A138565 4 12
%e A138565 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.
%o A138565 (GAP)
%o A138565 Print("\n") ;
%o A138565 for o in [ 1 .. 40 ] do
%o A138565     n := NumberSmallGroups(o) ;
%o A138565     og := [] ;
%o A138565     for i in [1 .. n] do
%o A138565         g := SmallGroup(o,i) ;
%o A138565         if IsAbelian(g) then
%o A138565             H := AutomorphismGroup(g) ;
%o A138565             ho := Order(H) ;
%o A138565             Add(og,ho) ;
%o A138565         fi ;
%o A138565     od;
%o A138565     Sort(og) ;
%o A138565     Print(og) ;
%o A138565     Print("\n") ;
%o A138565 od; # _R. J. Mathar_, Jul 13 2013
%K A138565 easy,nonn,tabf
%O A138565 1,3
%A A138565 _Benoit Jubin_, May 12 2008