A133777 Number of isomorphism types of groups of order n!.

1, 1, 1, 2, 15, 47, 840, 4539

Offset

0,4

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.

Links

Table of n, a(n) for n=0..7.

H. U. Besche, B. Eick and E. A. O'Brien, Number of isomorphism types of finite groups of given order

GAP package GrpConst

Formula

a(n) = A000001(n!). - M. F. Hasler, Dec 12 2010

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 degrees and a reflection w.r.t. an angle bisector. - M. F. Hasler, Dec 12 2010

Crossrefs

Cf. A000001.

Sequence in context: A336209 A256328 A041719 * A350383 A025213 A362303

Adjacent sequences: A133774 A133775 A133776 * A133778 A133779 A133780

Keyword

hard,nonn,more

Author

Peter C. Heinig (algorithms(AT)gmx.de), Jan 02 2008

Extensions

a(7) from Eric M. Schmidt, Sep 15 2014

Status

approved