A181941 Numbers n such that some group of order n has a non-cyclic commutator group.

12, 18, 24, 32, 36, 48, 50, 54, 56, 60, 64, 72, 75, 80, 81, 84, 90, 96, 98, 100, 108, 112, 120, 126, 128, 132, 144, 147, 150, 156, 160, 162, 168, 180, 192, 196, 198, 200, 204, 216, 224, 225, 228, 234, 240, 242, 243, 250, 252, 256, 264, 270, 276, 280, 288, 294, 300, 306, 312, 320, 324, 336, 338, 342

Offset: 1

Author

R. J. Mathar, Apr 03 2012

Keywords

• nonn

Comments

The complementary sequence 1, 2, 3, 4... is much denser and contains all n such that each group of order n has a cyclic commutator group.

Let the factorization of n into powers of squarefree mutually coprime numbers n_1, n_2, n_3, n_4, n_5,... be n = n_1 *n_2^2 *n_3^3 *n_4^4 * n_5^5*..., see A051903.

Then the complementary sequence contains n of the form n = n_1*n_2^2*n_3^3*n_4^4 under the constraints:

(i) n_4=1 or n_4=2

(ii) gcd(n, psi(n_2^2*n_3^3*n_4^4)) =1 where psi(k) = abs(A153038(k)) .

Examples

			1) Does not contain 10 = 10*1*1*1 where n_4=1 and gcd(10,|A153038(1)|)=1.
Both groups of order 10 have cyclic commutator groups: D10 has C5 and C10 has E.
2) Contains 12 = 3*2^2 where n_4=1 and gcd(12,|A153038(4)|) >1.
The group A4 of order 12 has a commutator group C2 x C2 which is not cyclic.
3) Contains 18 = 2*3^2 where n_4=1 and gcd(18,|A153038(9)|) >1.
The group (C3 x C3) : C2 of order 18 has a commutator group C3 x C3 which is not cyclic. (Gap notation, SmallGroup(18,4), where the colon is the semidirect product)
4) Contains 24 = 3*1*2^3 where n_4=1 and gcd(24,|A153038(9)|) >1.
5) Contains 32 = 1*1*1*1*2^5 where n_5>1.
6) Contains 48 = 3*1*1*2^4 where n_4=2 and gcd(48,|A153038(16)|)>1.
		

Links

• G. Pazderski, Die Ordnungen, zu denen nur Gruppen mit gegebener Eigenschaft gehören, Archiv Math. 10 (1) (1959) 331.

Programs

Maple

nsq := proc(n)
    local f,L ;
    L := [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] ;
    if n = 1 then
        return L;
    else
        for f in ifactors(n)[2] do
            p := op(1,f) ;
            e := op(2,f) ;
            i := e ;
            L := subsop(i=op(i,L)*p^e,L) ;
        end do:
        return L ;
    end if;
end proc:
Pazdn4 := proc(L)
    if nops(L) <4 then
        1;
    else
        sqrt(sqrt(op(4,L))) ;
    end if;
end proc:
hihno1 := proc(L)
    i := 0 ;
    for j from 1 to nops(L) do
        if op(j,L) > 1 then
            i := j ;
        end if;
    end do:
    i ;
end proc:
for n from 1 to 600 do
    nf := nsq(n) ;
    n4 := Pazdn4(nf) ;
    psarg := op(2,nf)*op(3,nf)*op(4,nf) ;
    if  ( n4 =1 or n4 =2)  and gcd(n, abs(A153038(psarg))) = 1 and hihno1(nf) < 5 then
        ;
    else
        printf("%d,",n) ;
    end if;
end do:

Mathematica

A153038[n_] := If[n == 1, 1, x = 1; Do[p = f[[1]]; e = f[[2]]; x = x*Product[1 - p^s, {s, 1, e}], {f, FactorInteger[n]}]; x];
nsq[n_] := Module[{f, L}, L = Table[1, {44}]; If[n == 1, Return[L], Do[{p, e} = f; L[[e]] = L[[e]]*p^e, {f, FactorInteger[n]}]]; L];
Pazdn4[L_] := If[Length[L] < 4, 1, Sqrt[Sqrt[L[[4]]]]];
hihno1[L_] := Module[{i, j}, i = 0; For[j = 1, j <= Length[L], j++, If[L[[j]] > 1, i = j]]; i];
Reap[For[n = 1, n <= 600, n++, nf = nsq[n]; n4 = Pazdn4[nf]; psarg = nf[[2]] * nf[[3]] * nf[[4]]; If[(n4 == 1 || n4 == 2) && GCD[n, Abs[A153038[psarg]]] == 1 && hihno1[nf] < 5, , Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jun 07 2024, after R. J. Mathar *)