A060702 -id:A060702 - cp's OEIS frontend

A357900 Number of groups of order A060702(n) with trivial center.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 5, 1, 1, 1, 1, 3, 1, 1, 1, 1, 6, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 5, 2, 5, 1, 1, 5, 2, 1, 2, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 3, 1, 4, 1, 1, 4, 1, 1, 17, 1, 1, 5, 1, 1, 1, 1, 8, 1, 1, 2, 1, 11, 1, 2, 2, 5, 1, 1, 1, 2, 1, 1, 3, 1, 1, 19

Offset: 1

Jianing Song, Oct 19 2022

Among the data currently known, it seems that the indices of records are n's such that A060702(n) = 1, 18, 54, 72, 162, 216, 486, 648, 972, 1458, ... with record values 1, 2, 5, 6, 17, 19, 72, 79, 109, 443, ...

			a(2) = 1 since there is a unique group of order A060702(2) = 6 with trivial center: S3.

Jianing Song, Table of n, a(n) for n = 1..372
Jianing Song, Number of centerless groups of order n <= 2022 (skipping n = 768, 1152, 1280, 1536, 1728, 1792, 1920, 2016)

Cf. A060702, A059806, A056867.

IsNilpotentNumber := function(n) # if n > 1 is a nilpotent number, then no group of order n has trivial center; see also A056867
    local c, omega, i, j;
    c := PrimePowersInt( n );
    omega := Length(c)/2;
    for i in [1..omega] do
        for j in [1..c[2*i]] do
            if GcdInt(n, c[2*i-1]^j-1) > 1 then
                return false;
            fi;
        od;
    od;
    return true;
end;
CountTrivialCenter := function(n) # returns the number of groups of order n with trivial center
    local count, i;
    if n > 1 and IsNilpotentNumber(n) then
        return 0;
    fi;
    count := 0;
    for i in [1..NumberSmallGroups(n)] do
        if(Size(Center(SmallGroup(n, i))) = 1) then
            count:=count+1;
        fi;
    od;
    return count;
end;

A059806 Minimal size of the center of G where G is a finite group of order n.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 7, 2, 9, 1, 11, 1, 13, 1, 15, 2, 17, 1, 19, 1, 1, 1, 23, 1, 25, 1, 3, 2, 29, 1, 31, 2, 33, 1, 35, 1, 37, 1, 1, 2, 41, 1, 43, 2, 45, 1, 47, 1, 49, 1, 51, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 3, 2, 65, 1, 67, 1, 69, 1, 71, 1, 73, 1, 1, 2, 77, 1

Offset: 1

Noam Katz (noamkj(AT)hotmail.com), Feb 24 2001

nonn

a(n) = n if and only if n belongs to sequence A051532 - Avi Peretz (njk(AT)netvision.net.il), Feb 27 2001

a(n) = 1 if and only if n occurs in A060702. - Eric M. Schmidt, Aug 27 2012

			a(6) = 1 because the symmetric group S_3 has trivial center.

Eric M. Schmidt, Table of n, a(n) for n = 1..2000
MathOverflow, Center of p-groups

Cf. A059807, A051532.

A059806 := function(n) local min, fact, i; if (n mod 6 = 0) then return 1; fi; if (IsPrimePowerInt(n)) then fact := Factors(n); if (Length(fact) <> 2) then return fact[1]; fi; fi; min := n; for i in [1..NumberSmallGroups(n)] do min := Minimum(min, Size(Center(SmallGroup(n, i)))); if (min = 1) then break; fi; od; return min; end; # Eric M. Schmidt, Aug 27 2012

For prime p and m >= 3, a(p^m) = p. - Eric M. Schmidt, Aug 27 2012

More terms from Eric M. Schmidt, Aug 27 2012

A340519 Smallest order of a non-abelian group with a center of order n.

Original entry on oeis.org

6, 8, 18, 16, 30, 24, 42, 32, 54, 40, 66, 48, 78, 56, 90, 64, 102, 72, 114, 80, 126, 88, 138, 96, 150, 104, 162, 112, 174, 120, 186, 128, 198, 136, 210, 144, 222, 152, 234, 160, 246, 168, 258, 176, 270, 184, 282, 192, 294, 200, 306, 208, 318, 216, 330, 224, 342, 232, 354, 240, 366, 248

Offset: 1

Bob Heffernan and Des MacHale, Jan 24 2021; corrected Feb 14 2021

nonn

a(n) is 6n if n is odd and 4n if n is even. This is because the groups involved are C(n) X S3 if n is odd, where S3 is the symmetric group of order 6, and C(n/2) X D8 if n is even, where D8 is the dihedral group of order 8 and C(m) is the cyclic group of order m.

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

Equals 2*A106833.

Cf. A059806, A060702, A340518.

Table[If[OddQ[n],6n,4n],{n,100}] (* Harvey P. Dale, Mar 03 2023 *)

A357924 Number of groups of order n with trivial center.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 5, 1, 1, 1, 1, 0, 3, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 6, 0, 1, 1, 0, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 5, 0, 2, 0, 5

Offset: 1

Jianing Song, Oct 20 2022

A357900 is the main sequence.

Cf. A357900, A060702 (indices of terms > 0).

cp's OEIS Frontend

A357900 Number of groups of order A060702(n) with trivial center.

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A059806 Minimal size of the center of G where G is a finite group of order n.

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A340519 Smallest order of a non-abelian group with a center of order n.

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A357924 Number of groups of order n with trivial center.

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