Isomorphism and Generators in Z sub P

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SUMMARY

The discussion centers on proving that the automorphism group Aut(Z_p) is isomorphic to the cyclic group Z_{p-1}. Participants emphasize that an automorphism must preserve group structure, being both one-to-one and onto. The key insight is that knowing the image of a generator x under an automorphism f allows one to determine the image of all elements in Z_p. Furthermore, it is established that if x is a generator, then f(x) must also be a generator, confirming the isomorphism with the group of units of Z_p.

PREREQUISITES
  • Understanding of group theory concepts, specifically automorphisms.
  • Familiarity with cyclic groups and their properties.
  • Knowledge of the structure of the integers modulo a prime, Z_p.
  • Basic comprehension of isomorphisms in algebra.
NEXT STEPS
  • Study the properties of automorphisms in group theory.
  • Learn about the structure of the group of units in modular arithmetic.
  • Explore cyclic groups and their generators in detail.
  • Investigate the proof techniques for establishing isomorphisms between groups.
USEFUL FOR

This discussion is beneficial for students and researchers in abstract algebra, particularly those studying group theory, automorphisms, and cyclic groups. It is especially relevant for individuals tackling advanced topics in algebraic structures.

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Homework Statement


Let P be a prime integer, prove that Aut(Z sub P) ≈ Z sub p-1


Homework Equations



none

The Attempt at a Solution


groups must preserve the operation, be 1-1, and be onto and they can be called an isomorphism. Z sub p-1 has one less element in it so and all the elements in them are the same except for the one less element. Not sure what this tells me though. HELP ME FOR LINEARRRRR
 
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Let f be an automorphism of Z_p, and x a generator for Z_p, so that <x>=Z_p. Explain why f is determined by what it does to x, i.e. why knowing f(x) suffices to to know where f sends any other element of the group.

Now think about whether (x is a generator) => (f(x) is a generator) is true. You will see that Aut(Z_p) is isomorphic to the group of units of Z_p -- there is a natural isomorphism. Think how you can prove that the latter is cyclic.
 

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