It is NOT true in general because consider Z6, the cyclic group of order 6, i.e. Z6 ={0,1,2,3,4,5} under addition. Notice that if 2x = 2y, it does not imply that x = y. For example, let x = 0 and y = 3. You'll see that 2x = 2*0= 0 and 2y = 2*3=6=0, but x=3≠0=y.
When you say "Z6, the cyclic group of order 6", you mean Z6 = {0,1,2,3,4,5} under addition modulo 6, correct? The set Z6 = {0,1,2,3,4,5} is not a group under multiplication modulo 6 (0 does not have a multiplicative inverse modulo 6), which appears to be the operation you are using in your comment.
It is NOT true in general because consider Z6, the cyclic group of order 6, i.e. Z6 ={0,1,2,3,4,5} under addition. Notice that if 2x = 2y, it does not imply that x = y. For example, let x = 0 and y = 3. You'll see that 2x = 2*0= 0 and 2y = 2*3=6=0, but x=3≠0=y.
In general, cyclic group Zn has the left and right cancellation property iff n is prime.
When you say "Z6, the cyclic group of order 6", you mean Z6 = {0,1,2,3,4,5} under addition modulo 6, correct?
The set Z6 = {0,1,2,3,4,5} is not a group under multiplication modulo 6 (0 does not have a multiplicative inverse modulo 6), which appears to be the operation you are using in your comment.
@@iliekmathphysics Correct!