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A133777
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Number of isomorphism types of groups of order n!.
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0
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OFFSET
| 0,4
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COMMENTS
| This sequence is interesting in view of Cayley's theorem which says that every finite group with n elements is isomorphic to a subgroup of the symmetric group S_n whose number of elements is n!. Therefore a(n) - 1 gives the number of groups "competing" with S_n in this respect. The eighth term, a(7), i.e. the number of isomorphism types of groups of order 7!=5040, seems to be unknown.
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LINKS
| Hans Ulrich Besche, Number of isomorphism types of finite groups of given order
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FORMULA
| a(n)=A000001(n!). - M. F. Hasler, Dec 12 2010
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EXAMPLE
| a(0)=a(1)=1 because 0!=1!=1 and there is exactly one group of order one up to isomorphism.
a(2)=1 because there is exactly one group of order 2!=2, G={e,a} with a*a=e.
a(3)=2 because there are 2 groups of order 3!=6, namely the cyclic group Z/6Z and the nonabelian dihedral group of isometries of the triangle, generated by a rotation of 120° and a reflection w.r.t. an angle bisector. - M. F. Hasler, Dec 12 2010
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CROSSREFS
| Cf. A000001.
Sequence in context: A152015 A162256 A041719 * A025213 A116693 A154565
Adjacent sequences: A133774 A133775 A133776 * A133778 A133779 A133780
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KEYWORD
| hard,nonn,more
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AUTHOR
| Peter C. Heinig (algorithms(AT)gmx.de), Jan 02 2008
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