determine the centralizer and the order of the

Question

Determine the centralizer and the order of the conjugacy:

1)      Determine the centralizer and the order of the conjugacy class of the matrix [1, 1; 0, 1] in Gl­₂(F₃).

2)      Prove that the icosahedral group has a subgroup of order 10.

3)      Let G be a group of order n which acts nontrivially on a set of order r. Prove that if n>r!, then G has a proper normal subgroup.

4)      Determine the conjugacy classes in the dihedral group D₁₀ and write down the class equation for this group. Find the center of D₁₀.