Abstract Algebra Proof question

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SUMMARY

The discussion centers on proving that the subset A of Sn, consisting of permutations that can be expressed as a product of an even number of transpositions, forms a group under permutation multiplication. The user suggests starting with specific permutations alpha and beta and exploring closure by examining their product. The conclusion is that the product of two permutations in A also belongs to A, thus satisfying one of the group axioms. The proof requires verifying all group axioms, particularly the significance of the even number of transpositions.

PREREQUISITES
  • Understanding of group theory and its axioms
  • Familiarity with permutations and transpositions
  • Knowledge of the symmetric group Sn
  • Ability to manipulate algebraic expressions involving permutations
NEXT STEPS
  • Study the properties of symmetric groups, specifically Sn
  • Learn about group axioms and how they apply to permutation groups
  • Explore the concept of transpositions and their role in permutation parity
  • Investigate closure properties in algebraic structures
USEFUL FOR

Students of abstract algebra, mathematicians focusing on group theory, and anyone interested in the properties of permutations and their applications in mathematical proofs.

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Homework Statement


The question is:
Let A be a subset of Sn that contains all permutations alpha such that alpha can be written as a product of an even number of transpositions. Prove that A is a group with product of permutations.

I understand what I need to do to prove it, but I am not sure how to start it. Do I use:
Let alpha=(a1a2a3...an) and
beta=(b1b2b3...bn) and try to find closure,

alphabeta=(a1a2a3...an)(b1b2b3...bn)=(a1a2)(a1a3)...(a1an-1)(a1an)(b1b2)(b1b3)...(b1bn-1)(b1bn),

or am I going about it the wrong way?


Homework Equations





The Attempt at a Solution

 
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What is a group? There are some axioms A must satisfy in order for it to be called one. For instance, the product of two things in A must also lie in A. Translating this to the situation at hand: if two permutations can be written as a product of an even number of transpositions, can their product be as well? Obviously yes! Now check the other axioms similarly.
 
Where did you use the fact that they can be written as an even number of transpositions?
 

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