A particular subgroup of a Free Group is normal

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SUMMARY

In the discussion, it is established that for a free group F and a subgroup N generated by the set {x^n : x is in F and n is fixed}, the subgroup N is normal in F. The key insight provided is the manipulation of the expression yx^ny^{-1} to show that it equals (yxy^{-1})^n, confirming the normality of N in F. This result is crucial for understanding the structure of free groups and their subgroups.

PREREQUISITES
  • Understanding of free groups in group theory
  • Familiarity with subgroup properties and normal subgroups
  • Basic knowledge of group homomorphisms
  • Experience with algebraic manipulation of group elements
NEXT STEPS
  • Study the properties of normal subgroups in group theory
  • Explore the concept of free groups in more depth
  • Learn about group actions and their implications on subgroup normality
  • Investigate examples of free groups and their normal subgroups
USEFUL FOR

Mathematicians, particularly those specializing in algebra and group theory, as well as students studying advanced topics in abstract algebra.

empyreandance
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Hello friends,

I'm working through my book and I'm having a lot of trouble coming to terms / believing this. Could anyone assist?

Let F be a free group and N be the subgroup generated by the set {x^n : x is in F and n is fixed} then N is normal in F.

Any ideas?
 
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Use this trick:

yx^ny^{-1}=(yxy^{-1})^n
 
You saved me yet again! Thank you friend.
 

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