A059807 -id:A059807 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

Des MacHale, Jan 24 2021

nonn

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

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

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.

Cf. A059807, A060793, A146992, A341293.

# 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

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

More terms from Miles Englezou, Feb 26 2024

cp's OEIS Frontend

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

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