cp's OEIS Frontend

A340518 Smallest order of a finite group with a commutator subgroup of order n.

%I A340518 #47 Mar 28 2024 23:51:05
%S A340518 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 A340518 50,104,54,84,58,120,62,96,66,136,70,72,74,152,78,160,82,168,86,176,
%U A340518 90,184,94,96,98,200,102,156,106,216,110,168,114,232
%N A340518 Smallest order of a finite group with a commutator subgroup of order n.
%C A340518 By Lagrange's Theorem a(n) is a multiple of n.
%C A340518 Are all terms after the first even?
%C A340518 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 A340518 Miles Englezou, <a href="/A340518/b340518.txt">Table of n, a(n) for n = 1..255</a>
%H A340518 Groupprops, <a href="https://groupprops.subwiki.org/wiki/Subgroup_structure_of_dihedral_groups">Subgroup structure of dihedral groups</a>.
%H A340518 Miles Englezou, <a href="/A340518/a340518.txt">Proof that A340518(n) is even</a>.
%F A340518 a(2n+1) = 4n+2. - _Miles Englezou_, Mar 08 2024
%e A340518 The fourth term is 12, because 12 is the smallest order of a group G with |G'| =  4, A_4 being an example.
%o A340518 (GAP)
%o A340518 # Produces a list A of the first 255 terms
%o A340518 A:=[];
%o A340518 N:=[1..255];
%o A340518 F:=[1..20];     # for large n the array F may need to be extended beyond 20
%o A340518 for n in N do
%o A340518     for k in F do
%o A340518     L:=List([1..NrSmallGroups(n*k)],i->Size(DerivedSubgroup(SmallGroup(n*k,i))));;
%o A340518     if Positions(L,n)<>[] then
%o A340518         Add(A,n*k);
%o A340518         break;
%o A340518     fi;
%o A340518     od;
%o A340518 od; # _Miles Englezou_, Feb 26 2024
%Y A340518 Cf. A059807, A060793, A146992, A341293.
%K A340518 nonn
%O A340518 1,2
%A A340518 _Des MacHale_, Jan 24 2021
%E A340518 More terms from _Miles Englezou_, Feb 26 2024