Subgroups of Symmetric and Dihedral groups

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SUMMARY

This discussion focuses on the challenges of working with subgroups of symmetric and dihedral groups, particularly in demonstrating whether a subgroup is normal, finding subgroups of specific orders, and proving the non-existence of such subgroups. The user expresses difficulty in applying these concepts effectively and seeks strategies and exercises to enhance their understanding. The discussion highlights the need for targeted practice and resources to master subgroup theory within these group types.

PREREQUISITES
  • Understanding of group theory concepts, specifically symmetric and dihedral groups.
  • Familiarity with subgroup properties, including normal subgroups.
  • Knowledge of permutation operations and their applications in group theory.
  • Experience with mathematical proofs and counterexamples in abstract algebra.
NEXT STEPS
  • Study the criteria for normal subgroups in symmetric groups.
  • Explore the Sylow theorems for finding subgroups of specific orders.
  • Practice exercises involving the construction and analysis of dihedral groups.
  • Review advanced group theory textbooks for additional exercises and examples.
USEFUL FOR

Students of abstract algebra, particularly those preparing for exams in group theory, as well as educators seeking to provide effective exercises on symmetric and dihedral groups.

Avatrin
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I am having problem working with the objects in the title. Working with permutations, rotations and reflections is fine, but I have problem with the following:

Showing a subgroup is or is not normal (usually worse in the case of symmetric groups)

Finding a subgroup of order n.

Showing that there is no subgroup of order n.

I cannot remember encountering many exercises that helped me learn to work with subgroups of symmetric and dihedral groups. Are there any strategies I can follow, and, even better, any sets of exercises anybody here recommends?
 
My exams are over, and luckily, I didn't need much about this topic. Also, I guess my questions above were too broad.
 

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