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Math proof problem help

  1. Oct 9, 2009 #1
    #1 a) If ex = x for some elements e,x belong to S, we say e is a left identity for x; similarly, if xe = x we say e is a right identity for x. Prove that an element is a left identity for one element of S if and only if it is a left identity for every element of S. Let S be a non-empty set with a binary operation which is associative and both left and right transitive

    b) Prove that S has a unique identity element

    c) Deduce that S is a group under the given binary operation



    #2.Prove that n|φ(a^n-1) for every integer a≥2 and any positive integer n
     
    Last edited: Oct 10, 2009
  2. jcsd
  3. Oct 9, 2009 #2

    honestrosewater

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    How do you start? For (a), the second implication is immediate. Do you have other conditions for the operation? Is it associative? Do you have inverses? Is S the domain of a group?
     
  4. Oct 10, 2009 #3
    there exists a such that and xs1=y s2x=y..
     
  5. Oct 10, 2009 #4

    honestrosewater

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    Part of your post got lost. Is this an axiom?

    I missed your edit. What does it mean to be left and right transitive?
     
    Last edited: Oct 10, 2009
  6. Oct 10, 2009 #5
    thank you
    i have done this one

    but still no idea to do this one


    Prove that n|φ(a^n-1) for every integer a≥2 and any positive integer n
     
  7. Oct 11, 2009 #6

    honestrosewater

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    What does that formula say? What do "|", "φ", and "φ(x)" mean?
     
  8. Oct 12, 2009 #7

    MathematicalPhysicist

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    For 2, use the fact that:
    φ(n)=(p1^k1-p1^(k1-1))....(pr^kr-pr^kr+1)
    for n= p1^k1 .... pr^kr
    for pi primes, and because a^n-1=(a-1)(a^n-1+....+1)
    Now prove this theorem by induction.
     
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