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

1,2

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.

Links

Table of n, a(n) for n=1..60.

Formula

T(n,1) = n.

Example

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

Prog

(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;

Crossrefs

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

Sequence in context: A365739 A217714 A076965 * A375280 A113232 A095261

Adjacent sequences: A370691 A370692 A370693 * A370695 A370696 A370697

Keyword

nonn,tabf

Author

Miles Englezou, Feb 27 2024

Status

approved