Identity Property in Cayley Table

Question

I am working on a project which requires me to discern if the cayley tables in my text files, have identity, associative, inverse, and abelian properties.. I am currently working on the identity function, and while I believe I must use two nested for loops to cycle through the rows and columns of the tables. I am unable to find anything which may push me in the right direction, any help is appreciated. Thanks Jessica

Just wanted to add an update: This is what I finally came up with, posting just in case it could still use work. Many thanks.

group_el Group::getIdentity()
{
for (int i=0; i<order; i++)
   {
    bool identIsi = true;
    for (int j=0; j<order; j++)
       {
        if ((op(i,j)==i) && (op(j,i)==i))  //if i*j =i same as j*i = i then i is identity
          {
            return i; 
          }
        else 
          {
            identIsi = false;
          }
       }
    }

return NO_IDENTITY; }

Answer 1

C++ won't be the easiest language to work with, but it will be doable.

I took a class on abstract algebra a few years ago and from what I recall, you just have to write out the expression for every possible combination (since some groups are non-commutative you will need to do A*B and B*A, where * is an arbitrary operator) and work from there.

In order to decrease complexity, I'd do it in this order:

1. Scan for the identity element. There (should) be only one and once you determine its value the rest of the calculations should become a tad bit easier.
2. Scan for two elements that multiply to give the identity element. Those two elements (in the proper order) are inverses.
3. Determining whether the group is abelian will be simple. Just follow the definition of an abelian group.