cp's OEIS Frontend

A370694 Irregular array read by rows: T(n,k) is the number of endomorphisms of the k-th group of order n, with each row ordered so as to be nondecreasing.

%I A370694 #20 Mar 28 2024 23:52:06
%S A370694 1,2,3,4,16,5,6,10,7,8,28,32,36,512,9,81,10,26,11,12,20,33,48,64,13,
%T A370694 14,50,15,16,36,48,52,64,96,100,128,224,256,448,1024,1088,65536,17,18,
%U A370694 36,82,162,730,19,20,36,52,80,144,21,57,22,122,23,24
%N A370694 Irregular array read by rows: T(n,k) is the number of endomorphisms of the k-th group of order n, with each row ordered so as to be nondecreasing.
%C A370694 Unlike Aut(G), End(G) is, in general, not a group but a set. However, when G is an abelian group, End(G) is a ring.
%C A370694 If s is the largest k of a row n, T(p^r,s) = p^(r^2). This corresponds to the elementary abelian group G of order p^r, which is isomorphic to an r-dimensional vector space V over the finite field of characteristic p. As every group endomorphism of G is equivalent to a linear transformation of V, and every linear transformation is an r X r matrix with each entry ranging over p possible values, there are therefore p^(r^2) unique matrices, and consequently p^(r^2) endomorphisms of G.
%F A370694 T(n,1) = n.
%e A370694 First 16 rows:
%e A370694  1
%e A370694  2
%e A370694  3
%e A370694  4  16
%e A370694  5
%e A370694  6  10
%e A370694  7
%e A370694  8  28  32  36  512
%e A370694  9  81
%e A370694 10  26
%e A370694 11
%e A370694 12  20  33  48   64
%e A370694 13
%e A370694 14  50
%e A370694 15
%e A370694 16  36  48  52  64  96  100  128  224  256  448  1024  1088  65536
%o A370694 (GAP)
%o A370694 # Produces the terms of the first 31 rows.
%o A370694 LoadPackage("sonata");; # the sonata package needs to be loaded to call the function Endomorphisms. Sonata is included in the latest versions of GAP.
%o A370694 A:=[];;
%o A370694 B:=[];;
%o A370694 for n in [1..31] do
%o A370694     for i in [1..NrSmallGroups(n)] do
%o A370694         Add(B,Size(Endomorphisms(SmallGroup(n,i))));
%o A370694     od;
%o A370694     for k in [1..Size(SortedList(B))] do
%o A370694         Add(A,SortedList(B)[k]);
%o A370694     od;
%o A370694     B:=[];
%o A370694 od;
%Y A370694 Cf. A137316 (number of automorphisms of (n,k)).
%K A370694 nonn,tabf
%O A370694 1,2
%A A370694 _Miles Englezou_, Feb 27 2024