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A340518
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Smallest order of a finite group with a commutator subgroup of order n.
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3
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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The fourth term is 12, because 12 is the smallest order of a group G with |G'| = 4, A_4 being an example.
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PROG
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(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;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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