Group homomorphisms between cyclic groups

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SUMMARY

This discussion focuses on group homomorphisms between cyclic groups, specifically the mapping \(\phi: C_4 \to C_6\). The key definition of a homomorphism is established as \(\phi(a*b) = \phi(a) * \phi(b)\), which also implies that inverses are preserved. A critical insight is that specifying \(\phi(1)\) determines the values of \(\phi\) for all elements in \(C_4\). Additionally, it is confirmed that the order of \(\phi(g)\) must divide the order of \(g\), reinforcing the structural integrity of group homomorphisms.

PREREQUISITES
  • Understanding of cyclic groups, specifically \(C_4\) and \(C_6\)
  • Familiarity with the definition and properties of group homomorphisms
  • Knowledge of group order and its implications
  • Basic algebraic manipulation and proof techniques
NEXT STEPS
  • Study the properties of cyclic groups and their homomorphisms in detail
  • Learn how to prove that the order of \(\phi(g)\) divides the order of \(g\)
  • Explore examples of homomorphisms between different cyclic groups
  • Investigate the implications of specifying \(\phi(1)\) on the mapping of other elements
USEFUL FOR

This discussion is beneficial for students of abstract algebra, particularly those studying group theory, as well as educators looking for clear examples of group homomorphisms between cyclic groups.

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Describe al group homomorphisms \phi : C_4 --> C_6

The book I study from seems to pass over Group Homomorphisms very fast. So I decided to look at Artin's to help and it uses the same definition. So I think I am just not digesting something I should be. I know it's defined as \phi (a*b) = \phi (a) * \phi and that it maps the inverses to the inverses but I just have no idea how to apply these.
 
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I forgot a b in the definition of phi(a*b) = phi(a)*phi(b)
 
If you specify \phi(1), then what does this say about the value of \phi at the other elements in C_4? Also, a general fact about homomorphisms is that the order of \phi(g) must divide the order of g. Can you prove this? By the way, you can edit posts.
 

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