login
Number of isomorphism classes of polycyclic groups (or solvable groups) of order n.
1

%I #27 Nov 03 2017 03:43:35

%S 1,1,1,2,1,2,1,5,2,2,1,5,1,2,1,14,1,5,1,5,2,2,1,15,2,2,5,4,1,4,1,51,1,

%T 2,1,14,1,2,2,14,1,6,1,4,2,2,1,52,2,5,1,5,1,15,2,13,2,2,1,12,1,2,4,

%U 267,1,4,1,5,1,4,1,50,1,2,3,4,1,6,1,52,15,2,1

%N Number of isomorphism classes of polycyclic groups (or solvable groups) of order n.

%C For finite groups solvable is equivalent to polycyclic.

%H Muniru A Asiru, <a href="/A201733/b201733.txt">Table of n, a(n) for n = 1..500</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Polycyclic_group">Polycyclic group</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Solvable_groups">Solvable group</a>

%F a(n) = A000001(n) for n < 60.

%F a(n) <= A000001(n) with equality if and only if n is not in A056866. In particular a(n) = A000001(n) for odd n (this is the Feit-Thompson theorem). - _Benoit Jubin_, Mar 30 2012

%o (GAP)

%o a:=[];;

%o N:=120;;

%o for n in [1..N] do

%o a[n]:=0;;

%o for j in [1..NrSmallGroups(n)] do

%o if IsPcGroup(SmallGroup(n,j)) = true then

%o a[n]:=a[n]+1;

%o fi;

%o od;

%o Print(a[n],",");

%o od;

%K nonn

%O 1,4

%A _W. Edwin Clark_, Dec 04 2011