Proving Isomorphic Groups U(5) and U(10)

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The discussion centers on proving that the groups U(5) and U(10) are isomorphic. U(n) represents the group of integers less than n that are coprime to n, with multiplication modulo n. It is established that both U(5) and U(10) are cyclic groups of the same order, which confirms their isomorphism. The participants suggest using Cayley tables to demonstrate the cyclic nature of these groups, although it is noted that this may be unnecessary for the proof.

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  • Understanding of group theory concepts, specifically cyclic groups.
  • Familiarity with the definition and properties of U(n) groups.
  • Knowledge of modular arithmetic and multiplication modulo n.
  • Ability to construct and interpret Cayley tables.
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  • Research the properties of cyclic groups and their isomorphisms.
  • Learn how to explicitly compute U(n) for various integers n.
  • Study the construction and interpretation of Cayley tables in group theory.
  • Explore additional examples of isomorphic groups beyond U(5) and U(10).
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Students of abstract algebra, particularly those studying group theory, as well as educators and anyone interested in understanding the properties of cyclic groups and their isomorphisms.

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Homework Statement



For any positive integern, let U(n) be the group of all positive integers less than n and relatively prime to n, under multiplication modulo n. Show the the Groups U(5) and u(10) are isomorphic

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The Attempt at a Solution



any 2 cyclic groups of the same size have to be isomorphic.
For the answer to this problem should i do out a caley table for both groups to show they are cyclic? is this enough along with my statement? i guess what I am saying is...does the question ask me to prove that 2 cyclic groups of the same size are isomorphic?
 
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No, a Cayley table would be overkill! (although it would work).

I suggest you first work out what U(5) and U(10) is explicitely. Try to prove that they have the same order.
Then try to find a cyclic element in both groups.
 

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