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Fix a group G. We say that elements g1, g2∈G are conjugate if there exists h∈G such that hg1h−1 = g2. Prove...

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Advance Math

Fix a group G. We say that elementsg1,g2∈G are conjugateif there exists h∈Gsuch that

hg1h−1= g2.

  1. Prove that conjugacy is an equivalence relation.
  2. Prove that ifg∈Z(G),the center of G, then its conjugacy classes hascardinality one.
  3. Let G =Sn. Prove thath(i1i2...it)h−1  =(h(i1)h(i2) ...h(it)),where ij∈{1, 2, ..., n }.
  4. Prove that the partition of S3into conjugacy classes is {{e}, {(1 2), (23), (1 3)} , {(1 23), (1 3 2)}} .That is, thereare three distinct conjugacy classes: the set consisting of the1-cycle e is one class, the set of 2-cycles isanother class, and the set of 3-cycles forms the last conjugacyclass.
  5. Describe (with justification) the partition ofS4  into conjugacy classesexplicitly. Be sure to be clear as to exactly how many conjugacyclasses there are, give a representative element of each, and tellus how to determine which conjugacy class a given element ofS4 belongs. [Hint: You might want toinvent a concept of \"cycle type\" to describe your answer.]
  6. Are the elements
    1  2  3  
    02-7
    005
    and
    1  0  0  
    05Ï€
    -102
    conjugate in the group GL3(R)?Justify.

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