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Automorphism with order p

  1. Oct 15, 2008 #1
    1. The problem statement, all variables and given/known data
    Let A, B be groups and theta: A --> Aut(B) a homomorphism. For a in A denote theta(a)= theta_a in Aut(B). Equip the product set B x A={(b,a): a in A, b in B} with the binary operation (b,a)(b',a')= (b'',a'') where a''=aa' and b''=b(theta_a{b')).

    (a) Assume that p,q in N are prime and p divides (q-1). Consider the case A=Zmodp, B=Zmodq. Show that there exists phi in Aut(Zmodq) which has order p.

    Hint: Use Cauchy's Thm for Abelian groups

    (b) Deduce that for any 2 primes p,q in N such that p|(q-1) there is a non-Abelian group of order pq.

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    (a) If p divides q-1, then p=-1 in mod q. Don't really know where to go from there.....

    (b) No idea.
     
    Last edited: Oct 15, 2008
  2. jcsd
  3. Oct 16, 2008 #2

    morphism

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    (a) What is the order of Aut(B)?

    (b) Under the binary operation defined in the question, is BxA a group?
     
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