Interesting group theory problem

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SUMMARY

The discussion centers on a group theory problem involving the sets \{gag^{-1} | g \in G\} and G / C(a). Participants are tasked with demonstrating that these two sets have the same size by establishing a bijection. The hints provided focus on identifying a suitable candidate for this bijection, which is essential for solving parts b and c of the problem. The conversation emphasizes the importance of understanding the concept of the center of a group and its implications in group theory.

PREREQUISITES
  • Understanding of group theory concepts, specifically group actions and conjugation.
  • Familiarity with the center of a group, denoted as C(a).
  • Knowledge of bijections and their role in set theory.
  • Basic problem-solving skills in abstract algebra.
NEXT STEPS
  • Research the properties of group actions and their applications in group theory.
  • Study the concept of the center of a group and its significance in conjugacy classes.
  • Explore examples of bijections in set theory to strengthen understanding.
  • Practice solving similar group theory problems to enhance problem-solving skills.
USEFUL FOR

Students studying abstract algebra, particularly those focusing on group theory, as well as educators seeking to deepen their understanding of group actions and conjugacy.

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Homework Statement


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Homework Equations


center


The Attempt at a Solution


i can do part a.
can you give me hints on part b and c?
 

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For (b), you are asked to show that the sizes of these two sets are the same: [itex]\{gag^{-1} | g \in G\}[/itex] and [itex]G / C(a)[/itex]. The natural thing to do is to try to find a bijection between the two. There's only one obvious candidate that I can see.
 

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