A137316 -id:A137316 - cp's OEIS frontend

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.

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, 48, 18, 8, 24, 12, 10, 22, 8, 16, 336, 20, 480, 12, 18, 108, 11232, 12, 36, 28, 8, 30, 16, 32, 128, 384, 1536, 21504, 9999360, 20, 16, 24, 12, 36, 96, 288, 36, 18, 24, 16, 32, 672

Offset: 1

Benoit Jubin, May 12 2008

This is a subtable of A137316.
The length of the n-th row is A000688(n).
The largest value of the n-th row is A061350(n).
The number phi(n) = A000010(n) appears in the n-th row.
The number A064767(n) appears in the (n^3)-th row.
The number A062771(n) appears in the (2n)-th row.

			The table begins as follows:
1
1
2
2 6
4
2
6
4 8 168
6 48
4
10
4 12
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.

C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, arXiv:math/0605185 [math.GR], 2006.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, Amer. Math. Monthly 114 (2007), no 10, 917-923.
D. MacHale and R. Sheehy, Finite groups with few automorphisms, Math. Proc. Roy. Irish Acad., 104A(2) (2004), 231--238.

Print("\n") ;
for o in [ 1 .. 40 ] do
    n := NumberSmallGroups(o) ;
    og := [] ;
    for i in [1 .. n] do
        g := SmallGroup(o,i) ;
        if IsAbelian(g) then
            H := AutomorphismGroup(g) ;
            ho := Order(H) ;
            Add(og,ho) ;
        fi ;
    od;
    Sort(og) ;
    Print(og) ;
    Print("\n") ;
od; # R. J. Mathar, Jul 13 2013

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.

Original entry on oeis.org

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

Miles Englezou, Feb 27 2024

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.

			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

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

# 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;

T(n,1) = n.

A385480 Sum of the orders of every automorphism group of order n.

Original entry on oeis.org

1, 1, 2, 8, 4, 8, 6, 212, 54, 24, 10, 64, 12, 48, 8, 20936, 16, 552, 18, 132, 54, 120, 22, 856, 500, 168, 11844, 216, 28, 192, 30, 10047248, 20, 288, 24, 2856, 36, 360, 180, 2128, 40, 468, 42, 520, 216, 528, 46, 61696, 2058, 13080, 32, 876, 52, 320400, 150, 3960

Offset: 1

Miles Englezou, Jun 30 2025

nonn

Let G be a group of order n, let N = {1, 2, ..., n}, and let f: G -> N be a bijection whereby f(G) = I is an index set of G. An automorphism phi of G is a permutation of N via f(phi(G)). It is tempting to ask the question 'how many permutations of N obey the group laws?'. However this question is not well-defined since it would require there being a natural single choice of bijection for every group of order n, which in general does not exist. Enumerating permutations of N which are automorphisms for every isomorphism class G will therefore depend on the choice of bijection for G. a(n) is the upper bound for all such enumerations of permutations of size n since a(n) is either: the maximum enumeration when the choice of bijections ensures that all permutations are distinct; or a(n) is the enumeration including all multiplicities when the choice of bijections leads to permutations which are not distinct.

			a(3) = 2 since for the one group of order 3, C_3, |Aut(C_3)| = 2.
a(4) = 8 since for the two groups of order 4, C_4 and C_2 x C_2, |Aut(C_4)| + |Aut(C_2 x C_2)| = 2 + 6 = 8.
a(8) = 212 since for the five groups of order 8, the sum of the orders of their automorphism groups is 4 + 8 + 8 + 24 + 168 = 212.

Cf. A137316.

a := function(n)
local T, k;
T := [];
for k in [1..NrSmallGroups(n)] do
    T := Concatenation(T, [Order(AutomorphismGroup(SmallGroup(n,k)))]);
od;
return Sum(T);
end;

a(n) is the sum of the k-th row in A137316.

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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.

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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.

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A385480 Sum of the orders of every automorphism group of order n.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

GAP

Formula