Consider the group A5. (a) Compute a = (1,2, 3)(3, 4, 5) and ß = (1, 2, 3)(2, 3, 4). (b) List possible types of cyclic decompositions for elements in A5. (c) Show that for any distinct a, b, c E {1,2,3, 4, 5} there is y E A5 so that y(1,2, 3)y-1 = (a, b, c). (d) Deduce that if a normal subgroup N 4 A5 contains a 3-cycle (a, b, c), then N = A5. %3|

3. Let a and ẞ be permutations in S11 with a = (7 4 1 3) (28) (5) (9 6 10 11), β = ( 1 2 3 8 7 10 4 5 6 1 6 7 8 9 10 11 5 11 4 239 (a) Express ẞ as a composition of disjoint cycles. (b) Find the fixed points of ẞ². (c) Express a oẞ as a composition of disjoint cycles. (d) Find the orders of a and B. [1] [1] [1] [2] (e) Determine whether a is an even or odd permutation, explaining your rea- soning clearly. [2] (f) Find S11 such that Boy = a, showing your reasoning clearly. [3]

Let GL(2,11) be the group of all invertible 2 × 2 matrices with entries in Z₁1, with group operation given my matrix multiplication. Consider the following two matrices in this group (where an entry listed as k is shorthand for [k]11): 3 3 10 4- (1₂0). B- (B) A = B= 8 8 (ii) Consider the subset of GL(2,11) given by G= {Am B : m, n = Z}. Show that G is a subgroup of GL(2, 11).

Exercise 3.6. Determine whether the following groups are cyclic: (1) (Q,+) (2) (Z2, +2) x (Z2, +2) (3) (Z2, +2) x (Z3, +3).