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

1, 8, 6, 12, 10, 24, 14, 24, 18, 40, 22, 24, 26, 56, 30, 48, 34, 72, 38, 80, 42, 88, 46, 48, 50, 104, 54, 84, 58, 120, 62, 96, 66, 136, 70, 72, 74, 152, 78, 160, 82, 168, 86, 176, 90, 184, 94, 96, 98, 200, 102, 156, 106, 216, 110, 168, 114, 232

Offset: 1

Author

Des MacHale, Jan 24 2021

Keywords

• nonn

Comments

By Lagrange's Theorem a(n) is a multiple of n.

Are all terms after the first even?

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

Examples

			The fourth term is 12, because 12 is the smallest order of a group G with |G'| =  4, A_4 being an example.
		

Links

• Miles Englezou, Table of n, a(n) for n = 1..255
• Groupprops, Subgroup structure of dihedral groups.
• Miles Englezou, Proof that A340518(n) is even.

Crossrefs

Cf. A059807, A060793, A146992, A341293.

Programs

GAP

# Produces a list A of the first 255 terms
A:=[];
N:=[1..255];
F:=[1..20];     # for large n the array F may need to be extended beyond 20
for n in N do
    for k in F do
    L:=List([1..NrSmallGroups(n*k)],i->Size(DerivedSubgroup(SmallGroup(n*k,i))));;
    if Positions(L,n)<>[] then
        Add(A,n*k);
        break;
    fi;
    od;
od; # Miles Englezou, Feb 26 2024

Formula

a(2n+1) = 4n+2. - Miles Englezou, Mar 08 2024

Extensions

More terms from Miles Englezou, Feb 26 2024