Isomorphism and Cyclic Groups

  • Thread starter essie52
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  • #1
essie52
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Homework Statement


I need to prove that any isomorphism between two cyclic groups maps every generator to a generator.


2. The attempt at a solution
Here what I have so far:

Let G be a cyclic group with x as a generator and let G' be isomorphic to G. There is some isomorphism phi: G --> G'. Since phi is surjective then for any y in G' there exists some x in G such that phi(x) = y. Since x generates G then every element in x must be in the form of x^k for some integer k. Phi therefore, is determined by its value on x. The formula phi(x^k) = y^k defines the isomorphism.

This is the point where I go, "what now?" Any help appreciated! E

PS We have not discussed kernel in this class.
 

Answers and Replies

  • #2
micromass
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So you got two cyclic groups G and G' and an isomorphism f:G-->G'.
Let x be a generator of G, then every g in G can be written as xk=g.
You need to show that f(x) is a generator of G'. So pick an arbitrary h in G'. You'll need to find a k such that f(x)k=h...
 

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