Suppose n is as in the previous part. Find all the Sylow 2-supgroups of D_n. Describe them in terms of x and y.

Describe the set of primes p for which x² - 11 splits into linear factors over Zp. 

Let p be an odd prime that divides n, n = p^r l, p + 1. Suppose C = (x¹). Show that C is the unique Sylow p-subgroup of D_n.

Which of the following statements are true and which are false? Give reasons for your answer.

(a) If a finite group G acts on a finite set S, then G_s1 = G_s2 for all s₁, $₂ ∈ X.

(b) There are exactly 8 elements of order 3 in S₄.

(c)

(d)

(e) For any

Find where ?³ = 1, ? ≠ 1.

Suppose that is 2n × 2n matrix where A, B, C and D are nxn matrices. Show that M is symplectic if and only if the following conditions are satisfied:

A^tD - C^tB 1

A^tC - C^tA = 0

B^tD-D^tB = 0

(Hint: Use block matrix multiplication.)

Also, check that the matrix  where A is a n x n orthogonal matrix, is a symplectic matrix

Let . Show that G is the cyclic group of order six.