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

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SUMMARY

This discussion focuses on the construction of non-Abelian groups of order pq, where p and q are primes and p divides (q-1). The participants analyze a specific case using the groups A = Zmodp and B = Zmodq, employing the homomorphism theta: A --> Aut(B). Key conclusions include the existence of an automorphism phi in Aut(Zmodq) with order p, and the deduction that non-Abelian groups exist under the given conditions, leveraging Cauchy's Theorem for Abelian groups.

PREREQUISITES
  • Understanding of group theory concepts, specifically homomorphisms and automorphisms.
  • Familiarity with the structure of finite groups, particularly Abelian groups.
  • Knowledge of Cauchy's Theorem and its application in group theory.
  • Basic proficiency in modular arithmetic, especially with Zmodp and Zmodq.
NEXT STEPS
  • Study the properties of automorphisms in finite groups, focusing on Aut(Zmodq).
  • Explore Cauchy's Theorem in greater depth and its implications for group orders.
  • Research the classification of non-Abelian groups, particularly those of order pq.
  • Examine the implications of the binary operation defined on the product set B x A.
USEFUL FOR

This discussion is beneficial for mathematicians, particularly those specializing in abstract algebra, group theorists, and students studying advanced group theory concepts.

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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.

------

(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:
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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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