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A126103
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Number of pointed groups of order n: that is, Sum_{G = group of order n} Number of orbits in G under the full automorphism group of G.
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2
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1, 2, 2, 5, 2, 7, 2, 17, 5, 7, 2, 23, 2, 7, 4, 67, 2, 23, 2, 25, 8, 7, 2, 99, 5, 7, 18, 20, 2, 25, 2, 342, 4, 7, 4, 89, 2, 7, 8, 99, 2, 40, 2, 20, 10, 7, 2, 476, 5, 23, 4, 25, 2, 100, 10, 87, 8, 7, 2, 115, 2, 7, 24, 2602, 4, 25, 2, 25, 4, 25, 2, 461, 2, 7, 13, 20, 4, 40, 2, 504, 79, 7, 2, 141, 4, 7, 4, 83, 2, 83, 4, 20, 8, 7, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Number of pairs (G, g in G) for G a group of order n, g an orbit representative for action of Aut(G) on G.
This has the same relation to A000001 (groups) as A000081 (pointed trees, also called rooted trees) does to trees (A000055).
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LINKS
| Klaus Brockhaus, Table of n, a(n) for n=1..191
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PROG
| (MAGMA) D:=SmallGroupDatabase();
for o in [1..95] do
t1:=0;
t2:=NumberOfSmallGroups(D, o);
for n in [1..t2] do
G:=SmallGroup(D, o, n);
H:=AutomorphismGroup(G);
gg:=[];
for g in G do Append(~gg, g);
end for;
PH:=[];
for h in Generators(H) do
ph:=[];
for i in [1..#gg] do
j:=Position(gg, gg[i]@h);
Append(~ph, j);
end for;
Append(~PH, ph);
end for;
pH:=sub<SymmetricGroup(#gg) | PH>;
t1:=t1 + #Orbits(pH);
end for;
print(t1);
end for;
(MAGMA) D:=SmallGroupDatabase(); [ &+[ #Orbits(sub<SymmetricGroup(o) | [ [ Position(gg, h(gg[i])): i in [1..o] ] where gg is [g: g in G] : h in Generators(AutomorphismGroup(G)) ] where G is SmallGroup(D, o, n) > ) : n in [1..NumberOfSmallGroups(D, o)] ] : o in [1..95] ]; /* Klaus Brockhaus, Mar 08 2007 */
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CROSSREFS
| Cf. A000001 (groups). See A126102 for a different and somewhat inferior version.
Sequence in context: A059907 A024931 A029648 * A100030 A029603 A204202
Adjacent sequences: A126100 A126101 A126102 * A126104 A126105 A126106
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KEYWORD
| nonn
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AUTHOR
| Gabriele Nebe and N. J. A. Sloane (njas(AT)research.att.com), Mar 06 2007
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