I am a student in computer science - first year. I study linear linear algebra 2 - course of linear algebra 1. - In some institutions academic studies teach the courses together / teach in another way.
I tried to solve the question a few hours but I'm not sure how to solve it exactly.
the question is:
" 3. Consider U14 and U18. Are they isomorphic? Prove (find an isomorphism function) or disprove (find a reason why they cannot be). Remarks: (a) The set of integers of the group Un is the set of numbers from 1 to n − 1 which are relatively prime to n. For example, in U8 it is the set {1, 3, 5, 7}, in U12 it is the set {1, 5, 7, 11}, in U24 it is the set {1, 5, 7, 11, 13, 17, 19, 23}. When n is prime, it 1 is the set {1, 2, . . . , p − 1}. The operation is product modulo the n indicated as a subscript – that is, in the examples above – the 8 or 12 or 24. (b) In many textbooks, you may find the groups Un written as U(n).
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At first glance - you can think it is not isomorphic - because u14 - include only 5 prime which complement 14: 1+13 = 14 3+11 = 14 7 + 7 = 14 - whice i think its bad assume.
i think about the option 5 + 9 = 14 but 14 it not prime...
u17 include 6 primes - they do not have the same amount of organs - so they are not isomorphic
Am I right? And how do I prove it?
Hint: Check cardinality of each set and remember that the two sets have finite cardinality..
I just read your bottom line. Yes, just prove that if two sets are isomorphic and of finite cardinality, then they must have the same cardinality.