

A133777


Number of isomorphism types of groups of order n!.


0




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


CROSSREFS

Cf. A000001.
Sequence in context: A336209 A256328 A041719 * A025213 A290631 A116693
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



