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.

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, 14, 50, 15, 16, 36, 48, 52, 64, 96, 100, 128, 224, 256, 448, 1024, 1088, 65536, 17, 18, 36, 82, 162, 730, 19, 20, 36, 52, 80, 144, 21, 57, 22, 122, 23, 24

Offset: 1

Author

Miles Englezou, Feb 27 2024

Keywords

• nonn
• tabf

Comments

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.

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.

Examples

			First 16 rows:
 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
14  50
15
16  36  48  52  64  96  100  128  224  256  448  1024  1088  65536
		

Crossrefs

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

Programs

GAP

# Produces the terms of the first 31 rows.
LoadPackage("sonata");; # the sonata package needs to be loaded to call the function Endomorphisms. Sonata is included in the latest versions of GAP.
A:=[];;
B:=[];;
for n in [1..31] do
    for i in [1..NrSmallGroups(n)] do
        Add(B,Size(Endomorphisms(SmallGroup(n,i))));
    od;
    for k in [1..Size(SortedList(B))] do
        Add(A,SortedList(B)[k]);
    od;
    B:=[];
od;

Formula

T(n,1) = n.