How many subgroups does Z20 have List a generator for each

Step 1 of 3

We know that a set \(Z_{n}=\{0,1,2,3, \ldots ., n-1\}\) for \(n \geq 1\)   is a group under addition modulo n. So \(Z_{20}\) is the group over integers with the group operation as addition modulo 20.

Thus  \(Z_{20}=\{0,1,2,3, \ldots ., 19\}\) 

Also, the order of any subgroup of a group of order k must be a divisor of k. That is, the set \(< n \mid k >\) is the unique subgroup of \(Z_n\) of order \(k\).  

We have to find the number of subgroups of \(Z_{20}\). A set is a subgroup, if it satisfies the following properties:

-  It should be a subset

-  It should be closed

-  Identity element must be present

-  Inverse must be present.