A059806 -id:A059806 - cp's OEIS frontend

A060702 Orders of finite groups that have trivial center.

1, 6, 10, 12, 14, 18, 20, 21, 22, 24, 26, 30, 34, 36, 38, 39, 42, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 66, 68, 70, 72, 74, 75, 78, 80, 82, 84, 86, 90, 93, 94, 96, 98, 100, 102, 106, 108, 110, 111, 114, 116, 118, 120, 122, 126, 129, 130, 132, 134, 136

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 20 2001

nonn

Apart from the first element 1 this is a subsequence of A056868 because a nilpotent group has nontrivial center. If n = 0 mod 6 or n >= 6 and n = 2 mod 4 then n is in this sequence.
If n >= 6 and n == 2 mod 4 then n is a member of the sequence because of the dihedral group of order 2(2k+1). In addition, if p is a prime and p == 1 mod 4 then n=4p is a member of the sequence; however, if p == 3 mod 4 and p>5, then n=4p is not a member of the sequence. Furthermore, if n=pq where p and q are distinct odd primes with pDes MacHale and Mossie Crowe, Jul 05 2005
This sequence is closed under multiplication. - Eric M. Schmidt, Aug 27 2012

			The symmetric group S_3 of order 6 has trivial center so 6 belongs to the sequence.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A056868, A059806.

For the corresponding numbers of centerless groups of these orders see A357900.

The old entry 89 was an error, since it is a prime. - Robert F. Bailey (robertb(AT)math.carleton.ca) and Brett Stevens (brett(AT)math.carleton.ca), Jul 16 2009

Sequence extended and corrected by Eric M. Schmidt, Aug 27 2012

A059807 Maximal size of the commutator subgroup of G where G is a finite group of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 2, 1, 5, 1, 4, 1, 7, 1, 4, 1, 9, 1, 5, 7, 11, 1, 12, 1, 13, 3, 7, 1, 15, 1, 8, 1, 17, 1, 9, 1, 19, 13, 10, 1, 21, 1, 11, 1, 23, 1, 24, 1, 25, 1, 13, 1, 27, 11, 14, 19, 29, 1, 60, 1, 31, 7, 16, 1, 33, 1, 17, 1, 35, 1, 36, 1, 37, 25, 19, 1

Offset: 1

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

nonn

a(n) = 1 iff n belongs to sequence A051532. - Avi Peretz (njk(AT)netvision.net.il), Feb 27 2001

			a(6) = 3 because the commutator subgroup of the symmetric group S_3 is the group Z_3.

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

Cf. A059806, A051532.

A059807 := function(n) local max, fact, i; if (IsPrimePowerInt(n)) then fact := Factors(n); if (Length(fact) >= 2) then return n/fact[1]^2; fi; fi; max := 1; for i in [1..NumberSmallGroups(n)] do max := Maximum(max, Size(DerivedSubgroup(SmallGroup(n, i)))); od; return max; end; # Eric M. Schmidt, Sep 20 2012

For prime p and m >= 2, a(p^m) = p^(m - 2). - Eric M. Schmidt, Sep 20 2012

More terms from Eric M. Schmidt, Sep 20 2012

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

Original entry on oeis.org

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;

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 *)

cp's OEIS Frontend

A060702 Orders of finite groups that have trivial center.

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A059807 Maximal size of the commutator subgroup of G where G is a finite group of order n.

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

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GAP

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

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