Abstract Algebra: Groups and Subgroups

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SUMMARY

The discussion focuses on proving that the set H, defined as H = {a ∈ S | a * x = x * a for all x ∈ S}, is closed under the binary operation * given that * is associative. Participants clarify that the goal is to demonstrate that for any elements a, b in H, the product a * b also belongs to H. The solution hinges on the properties of associativity and the definition of H, negating the need for isomorphism in this context.

PREREQUISITES
  • Understanding of associative binary operations
  • Familiarity with group theory concepts
  • Knowledge of commutative properties in algebra
  • Basic experience with set notation and definitions
NEXT STEPS
  • Study the properties of groups in Abstract Algebra
  • Learn about closure properties in algebraic structures
  • Explore the concept of commutative groups
  • Review examples of binary operations and their applications
USEFUL FOR

Students of Abstract Algebra, particularly those preparing for exams involving group theory, as well as educators seeking to clarify concepts related to groups and subgroups.

taylor81792
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Homework Statement


The problem says: Suppose that * is an associative binary operation on a set S.
Let H= {a ε S l a * x = x * a for all x ε s}. Show that H is closed under *. ( We think of H as consisting of all elements of S that commute with every element in S)

My teacher is horrible so I am pretty lost in the class. I am aware of what the associative property is, but I'm not sure how to go about solving this question when it comes to the binary operation. This is going to be on my exam so I need to know how to solve it.
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Homework Equations


No relevant equations


The Attempt at a Solution


I know that with associative and groups you would try to prove its isomorphism but I'm not sure where to begin with this one
 
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No isomorphism required...

You're just trying to show that, for any a,b[itex]\in[/itex]H, a*b[itex]\in[/itex]H. You should be able to do this with your definition of H plus associativity.
 

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