If G is cyclic, and G is isomorphic to G', then G' is cycli

[Mr Davis 97]

Homework Statement

Title

Homework Equations

The Attempt at a Solution

This would seem to be very easy problem, since it's intuitively obvious that if two groups are isomorphic, and one is cyclic, then the other is cyclic too. However, I can't seem to formalize it with math.

Here is an idea. We can define that a group is cyclic by saying that for all b in G, there exists an a such that ##a^n = b## for some integer n. Now if G is cyclic, and we have an isomorphism ##\phi## from G to G', then it is true that ##\phi (a) ^n = \phi (b) = b'##, which means that G' is also cyclic. Does this sketch of a proof the right idea?

 

[fresh_42]

Homework Statement

Title

Homework Equations

The Attempt at a Solution

This would seem to be very easy problem, since it's intuitively obvious that if two groups are isomorphic, and one is cyclic, then the other is cyclic too. However, I can't seem to formalize it with math.

Here is an idea. We can define that a group is cyclic by saying that for all b in G, there exists an a such that ##a^n = b## for some integer n. Now if G is cyclic, and we have an isomorphism ##\phi## from G to G', then it is true that ##\phi (a) ^n = \phi (b) = b'##, which means that G' is also cyclic. Does this sketch of a proof the right idea?

Yes it is. You can leave out the ##b's## as you don't need them, the generator ##a## is sufficient, i.e. simply write ##a^n##. It doesn't need to be named. And you may forget the isomorphism, surjectivity is sufficient. And subgroups of cyclic groups are also cyclic.