Test by Bikram | Mathematics | Test 2 | Question: 26

Question

$O(G) = 12$ and $G$ is cyclic. The total number of generators possible in $G$ are __________.

Answer 1

The number of generators of a cyclic group of order O(G) is the number of  positive integers which are co-prime with O(G) and less than O(G) .

So in the given question, O(G) = 12. So, positive numbers less than 12 which are co-prime with 12 are 1, 5, 7, 11. So, if 'a' is the generator of the group then a^5, a^7, a^11 are also the generators of the group. So the answer is 4 generators.

For detailed discussion : https://math.stackexchange.com/questions/786452/how-to-find-a-generator-of-a-cyclic-group