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Mathematics 1214: Introduction to Group Theory Tutorial exercise sheet 2 1. Let ∗ be an operation on a set S. Suppose that e is an identity element for ∗, and that there are elements x, y1 , y2 ∈ S such that x ∗ y1 = e = x ∗ y2 , y1 ∗ x = e = y2 ∗ x and y1 6= y2 . (a) Explain why ∗ must fail to be associative. (b) Find the Cayley table of an operation with these properties on S = {e, f, g}. 2. Let ∗ be an associative operation on a set S, and suppose that e is an identity element for ∗. Show that if x, y ∈ S with x ∗ y = e and y ∗ x 6= e then x is not invertible with respect to ∗. [Hint: try a proof by contradiction.] 3. Let ∗ be an operation on a set S with an identity element e ∈ S, and let us write T = {x ∈ S : x 6= e}. Prove that ∗ is associative if and only if x ∗ (y ∗ z) = (x ∗ y) ∗ z for all x, y, z ∈ T . [Note: the statement “x, y, z ∈ T ” is shorthand for “x ∈ T , y ∈ T and z ∈ T ”.] 4. Let ∗ be an operation on a set S, and let ~ : S × S → S be given by x~y =y∗x (a) (b) (c) (d) for x, y ∈ S. Explain how the Cayley tables of ∗ and ~ are related. Prove that ∗ is commutative ⇐⇒ ∗ = ~ ⇐⇒ ~ is commutative. If (S, ∗) is a group, prove that (S, ~) is a group. Is the converse of (c) true? Either give a proof or a counterexample. [The converse of the statement “if P then Q” is “if Q then P ”.] 5. Given a group (G, ∗) and x, y ∈ G, consider the following two mappings: Lx : G → G, Lx (z) = x ∗ z and Ry : G → G, Ry (z) = z ∗ y for z ∈ G. (a) Prove that Lx ◦ Ry = Ry ◦ Lx . (b) Prove that (G, ∗) is abelian if and only if Lx = Rx for every x ∈ X. (c) Prove that the following statements are equivalent: (i). x = y (ii). Lx = Ly (iii). Rx = Ry [This means: show that any of these three statements is true if and only if any of the other statements is true. One way to do this is to prove that (i) ⇐⇒ (ii), and that (i) ⇐⇒ (iii).] 6. Which of the following is a group operation on M (2, R), the set of 2 × 2 matrices with real entries? As always, you should prove that your answers are correct. (a) matrix addition (b) matrix multiplication (c) the operation ∗ given by A ∗ B = AB − BA for matrices A and B