Question

In a cayley's table for a finite group, why does each element occur exactly once in each row and exactly once in each column?​

Answer 1

This can be proved by considering the opposite: suppose one row of a set's Cayley table did not contain a particular element. That is, let AX not equal B, whatever X is. ... Hence, each row and column of a group's Cayley table must contain exactly one of each element.