|
%I #47 Mar 28 2024 23:51:05
%S 1,8,6,12,10,24,14,24,18,40,22,24,26,56,30,48,34,72,38,80,42,88,46,48,
%T 50,104,54,84,58,120,62,96,66,136,70,72,74,152,78,160,82,168,86,176,
%U 90,184,94,96,98,200,102,156,106,216,110,168,114,232
%N Smallest order of a finite group with a commutator subgroup of order n.
%C By Lagrange's Theorem a(n) is a multiple of n.
%C Are all terms after the first even?
%C The above conjecture is true. For even n, a(n) is even by Lagrange's theorem. For odd n, it follows from the fact that every dihedral group D_{2n} has a commutator subgroup of order n when n is odd; as no group of odd order is perfect, 2*n is the smallest possible order that such a commutator subgroup can be contained in. (For an extended proof see the Miles Englezou link.) - _Miles Englezou_, Mar 08 2024
%H Miles Englezou, <a href="/A340518/b340518.txt">Table of n, a(n) for n = 1..255</a>
%H Groupprops, <a href="https://groupprops.subwiki.org/wiki/Subgroup_structure_of_dihedral_groups">Subgroup structure of dihedral groups</a>.
%H Miles Englezou, <a href="/A340518/a340518.txt">Proof that A340518(n) is even</a>.
%F a(2n+1) = 4n+2. - _Miles Englezou_, Mar 08 2024
%e The fourth term is 12, because 12 is the smallest order of a group G with |G'| = 4, A_4 being an example.
%o (GAP)
%o # Produces a list A of the first 255 terms
%o A:=[];
%o N:=[1..255];
%o F:=[1..20]; # for large n the array F may need to be extended beyond 20
%o for n in N do
%o for k in F do
%o L:=List([1..NrSmallGroups(n*k)],i->Size(DerivedSubgroup(SmallGroup(n*k,i))));;
%o if Positions(L,n)<>[] then
%o Add(A,n*k);
%o break;
%o fi;
%o od;
%o od; # _Miles Englezou_, Feb 26 2024
%Y Cf. A059807, A060793, A146992, A341293.
%K nonn
%O 1,2
%A _Des MacHale_, Jan 24 2021
%E More terms from _Miles Englezou_, Feb 26 2024
|
