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

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

1,3

Comments

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 the map f*phi*f^-1: N -> N. It is tempting to ask the question 'how many permutations of N are group automorphisms?'. However this question is not well-defined; to be answerable there would need to be a natural single choice of bijection for every group of order n, which in general does not exist. Any enumeration of permutations of N which are automorphisms for every isomorphism class G will therefore depend on the choice of bijection for every G. a(n) is an upper bound for any such enumeration 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.

Links

Table of n, a(n) for n=1..56.

Formula

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

Example

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.

Prog

(GAP)
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;

Crossrefs

Cf. A137316.

Sequence in context: A388527 A393779 A151928 * A365478 A271836 A349823

Adjacent sequences: A385477 A385478 A385479 * A385481 A385482 A385483

Keyword

nonn

Author

Miles Englezou, Jun 30 2025

Extensions

Name and comment clarified by Miles Englezou, Jan 08 2026

Status

approved