If isomorphic groups are regarded as being the same, prove, for each positive integer , that there are only finitely many distinct groups with exactly elements.
Thanks in advance.
another way: let and let be a group of order define multiplication in by where then is a group and
so every group of order defines a map note that if two groups define the same on then thus the number of groups of order is at
most the number of maps which can be defined from to which is
an easier way to get this upper bound (this just came to my mind!) is to look at the multiplication table of a group of order each of entries have at most pssibilities!!