In order to form the product of permutations 
we mean apply
first and then apply 
. In terms of the cycle representation
this means that products are carried out reading from left to right.
This assumes that you are now writing functions on the right!
To
carry out a product we construct cycles as described above but with one difference. Select
an element 
. The next element in the cycle is 
followed
by 
etc. Repeat this procedure until every element of A
appears in some cycle, again single cycles are suppressed.
To illustrate this, if 
and
then find and 
.
Note that the cycle in was written as
but could have been written as
.
Another way to carry out this multiplication process (although no different from what has already been said) is as follows. Write down your starting element. Read the cycles from left to right. If the starting element is in a cycle read its successor (you may have to wrap around to the beginning of the cycle). This is you new number. Now go to the next cycle in which this new number appears. Read its successor and this becomes the new number. Repeat until you reach the end of the cycles (you have reached the rightmost end). This is now written as the successor of the starting element. Now this number becomes the new starting element. Repeat the above to find its successor. When you obtain a successor which is the original starting element, close off your cycle. Open up a new cycle and start all over again until all numbers have appeared in exactly one cycle. Now erase cycles of length one.