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I need the right answer for Q18 with explanation.
1.13. EXERCISES 51 (d) However, prove that if X is of degree 1 and Y is irreducible, then so is XôY. 16. Construct the character table of S4. You may find the lifting process of Exercise 6 and the inner tensor products of Exercise 15 helpful. 17. Let Dn be the group of symmetries (rotations and reflections) of a regular n-gon. This group is called a dihedral group. (a) Show that the abstract group with generators p, T subject to the relations p" = T² = € and PT = TP-¹ is isomorphic to Dn. (b) Conclude that every element of Dn is uniquely expressible as Tip, where 0 ≤ i ≤ 1 and 0 ≤j≤n - 1. (c) Find the conjugacy classes of Dn. (d) Find all the inequivalent irreducible representations of Dn. Hint: Use the fact that Cn is a normal subgroup of Dn. 18. Show that induction is transitive as follows. Suppose we have groups G≥ H≥K and a matrix representation X of K. Then X1= (X1)1.