Abstract Algebra Question?
How many different commutative binary operations can be defined on a set of:
i) two elements
ii) three elements
iii) n elements
I can see the answers in the back of the book are 8, 729 and
n^[n(n+1)/2], respectively, but don’t know how they got them.
- Anonymous1 decade agoFavorite Answer
To count up the possibilities, it is useful to imagine a tabulation of the results of the operation on each possible pair of elements. First of all, since the operation is commutative, the table is symmetric with respect to its diagonal, so it is enough to specify the diagonal elements of the table plus the table cells below the diagonal. There are n*(n+1)/2 such cells, and n possibilities for the element appearing in each such cell, for a total of n^(n*(n+1)/2) different tables.