How Does Group Theory Reveal Non-Abelian Groups of Order pq?

[fk378]

Homework Statement

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.

 

[morphism]

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

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