cp's OEIS Frontend

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

%I A385480 #7 Jul 04 2025 01:24:57
%S A385480 1,1,2,8,4,8,6,212,54,24,10,64,12,48,8,20936,16,552,18,132,54,120,22,
%T A385480 856,500,168,11844,216,28,192,30,10047248,20,288,24,2856,36,360,180,
%U A385480 2128,40,468,42,520,216,528,46,61696,2058,13080,32,876,52,320400,150,3960
%N A385480 Sum of the orders of every automorphism group of order n.
%C A385480 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.
%F A385480 a(n) is the sum of the k-th row in A137316.
%e A385480 a(3) = 2 since for the one group of order 3, C_3, |Aut(C_3)| = 2.
%e A385480 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.
%e A385480 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.
%o A385480 (GAP)
%o A385480 a := function(n)
%o A385480 local T, k;
%o A385480 T := [];
%o A385480 for k in [1..NrSmallGroups(n)] do
%o A385480     T := Concatenation(T, [Order(AutomorphismGroup(SmallGroup(n,k)))]);
%o A385480 od;
%o A385480 return Sum(T);
%o A385480 end;
%Y A385480 Cf. A137316.
%K A385480 nonn
%O A385480 1,3
%A A385480 _Miles Englezou_, Jun 30 2025