a list all generators of G b How many subgroups does G have

a) list all generators of G

b) How many subgroups does G have? List generators for each of these subgroups.

a) The element will be the generator of the group if and only if its order is 18

The element is a generator of the group if and only if gcd(element,18) = 1

Hence the generators of the group Z18 are {1,3,5,7,11,13}

b) Z18 is a cyclic group, Z18 has the divisors equal to {1,2,3,6,9}

Hence there are five sub-groups of Z18

<0> is the sub-group of order 1

<1> is the sub-group of order 18 since it contains all the elements

<2> = {0,2,4,6,8,10,12,14,16} sub-group of order 9

<3> = {0,3,6,9,12,15} sub-group of order 6

<6> = {0,6,12} subgroup of order 3.

<9> = {0,9} subgroup of order 2.