A370694 - OEIS

%I #20 Mar 28 2024 23:52:06

%S 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 14,50,15,16,36,48,52,64,96,100,128,224,256,448,1024,1088,65536,17,18,

%U 36,82,162,730,19,20,36,52,80,144,21,57,22,122,23,24

%N 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 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 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 T(n,1) = n.

%e First 16 rows:

%e  1

%e  2

%e  3

%e  4  16

%e  5

%e  6  10

%e  7

%e  8  28  32  36  512

%e  9  81

%e 10  26

%e 11

%e 12  20  33  48   64

%e 13

%e 14  50

%e 15

%e 16  36  48  52  64  96  100  128  224  256  448  1024  1088  65536

%o (GAP)

%o # Produces the terms of the first 31 rows.

%o LoadPackage("sonata");; # the sonata package needs to be loaded to call the function Endomorphisms. Sonata is included in the latest versions of GAP.

%o A:=[];;

%o B:=[];;

%o for n in [1..31] do

%o     for i in [1..NrSmallGroups(n)] do

%o         Add(B,Size(Endomorphisms(SmallGroup(n,i))));

%o     od;

%o     for k in [1..Size(SortedList(B))] do

%o         Add(A,SortedList(B)[k]);

%o     od;

%o     B:=[];

%o od;

%Y Cf. A137316 (number of automorphisms of (n,k)).

%K nonn,tabf

%O 1,2

%A _Miles Englezou_, Feb 27 2024