Isomorphism between multiplicative modulo groups

Question

Group G = {1,2,3,4,5,6} multiplicative modulo 7 Group H = {1,5,7,11,13,17} multiplicative modulo 18

Show the groups are cyclic.

Found a generator 3 of G and 5 of H. How can I check quickly they are generators without computing all powers.

Provide a isomorphic between the groups

Identity goes to identity is obvious what about the rest? I was thinking 3 goes to 5 using (i)?

Yes, it clearly is: $U(7)$, the first group is cyclic by the duplicate. The second is $U(18)$, which is also cyclic with six elements by the duplicate, hence must be isomorphic. — Dietrich Burde, Jun 05 '16 at 16:25

score 1 · Answer 1 · answered Jun 05 '16 at 16:17

The isomorphism between two cyclic groups can be obtained by mapping the generator of the first cyclic group to the generator of the second cyclic group. In your case, define a map that takes element 3 to element 5, and since the map must be a homomorphism, it must take element $3^i$ to element $5^i$.

Isomorphism between multiplicative modulo groups

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