Group theory: Conjugates, commutativity

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SUMMARY

In the symmetric group S8, the number of conjugates for the permutation p = (1 3)(2 4 5 6) is calculated as 7! by dividing 8! by the product of the lengths of its cycles (4 and 2). For the permutation q = (1 3)(2 4)(7 6 5), the number of conjugates is determined to be 1680, derived from 8! divided by the product of the cycle lengths (2, 2, and 3). The discussion also addresses the need to find the size of the centralizer for both permutations to determine the number of commuting permutations.

PREREQUISITES
  • Understanding of symmetric groups, specifically S8
  • Knowledge of cycle notation in permutations
  • Familiarity with the concept of conjugates in group theory
  • Ability to compute factorials and their applications in combinatorial problems
NEXT STEPS
  • Study the concept of centralizers in group theory
  • Learn about the classification of permutations in symmetric groups
  • Explore the properties of conjugacy classes in S8
  • Investigate advanced topics in group theory, such as normal subgroups and their significance
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Mathematics students, particularly those studying abstract algebra and group theory, as well as educators looking for examples of conjugates and commutativity in symmetric groups.

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Homework Statement



In S8, consider p= (1 3)(2 4 5 6) and q=(1 3)(2 4)(7 6 5)

1. Find the number of conjugates of p and the number of conjugates of q
2. Find the number of permutations that commute with p and the number that commute with q

Homework Equations





The Attempt at a Solution



I think I've got myself really confused.

1. For p, we are working in S8 so I initally count 8!
I then divide by 2 as (1 3) = (3 1) and similarly divide by 4 because of the 4-cycle.
The disjoint cycles are not the same length so my final answer is 8! / (4 x 2) = 7!

For q, we have 8! / (2 x 2 x 3 x2) = 1680 where the final 2 is because we have 2 2-cycles.

2. I'm not so sure about how to do this. I think it is asking me to find the size of the centralizer of each element but I don't know how to do this.
 
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I think you need to divide by 2 again for p because 7 and 8 could either be mapped to themselves or to each other without it affecting anything else
 

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