Question

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

hg₁h⁻¹ = g₂.

1. Prove that conjugacy is an equivalence relation.
2. Prove that if g∈Z(G), the center of G, then its conjugacy classes has cardinality one.
3. Let G = S_n. Prove that h(i₁i₂ ... i_t)h⁻¹  = (h(i₁) h(i₂) ... h(i_t)), where i_j∈{1, 2, ... , n }.
4. Prove that the partition of S₃ into conjugacy classes is {{e} , {(1 2), (2 3), (1 3)} , {(1 2 3), (1 3 2)}} .That is, there are three distinct conjugacy classes: the set consisting of the 1-cycle e is one class, the set of 2-cycles is another class, and the set of 3-cycles forms the last conjugacy class.
5. Describe (with justification) the partition of S₄  into conjugacy classes explicitly. Be sure to be clear as to exactly how many conjugacy classes there are, give a representative element of each, and tell us how to determine which conjugacy class a given element of S₄ belongs. [Hint: You might want to invent a concept of "cycle type" to describe your answer.]
6. Are the elements

   1	2	3
   0	2	-7
   0	0	5

   and

   1	0	0
   0	5	π
   -1	0	2

   conjugate in the group GL₃(R)? Justify.

Answer 1