|
|
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
|
|
|
FORMULA
|
|
|
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
|
|
|
KEYWORD
|
hard,nonn,more
|
|
AUTHOR
|
Peter C. Heinig (algorithms(AT)gmx.de), Jan 02 2008
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|