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Mathematics 1214: Introduction to Group Theory Tutorial exercise sheet 4 1. The complex numbers C form a group under addition with identity element 0, such that the inverse of z ∈ C is −z. Consider the five sets N, Z, Q, R, C. Which of these are subgroups of (C, +)? 2. Find a subgroup H of (R, +) such that √ 2 ∈ H and Z ⊆ H 6= R. 3. Recall that An is the set of even permutations in Sn , which is a group under composition, and the order of this group is 21 |Sn |. (a) Show that A3 = {(1), (1 2 3), (1 3 2)}. (b) How many subsets of A3 are there? Which of these are subgroups? (c) What is the order of the group A4 ? List the elements of A4 , writing them as products of disjoint cycles. [We can write the identity permutation as (1)]. 4. Let (G, ∗) be a group with identity element e. For x ∈ G, let us write x0 = e and xk = |x ∗ x ∗{z· · · ∗ x} for k ∈ N. k times Suppose that m, n are non-negative integers. Prove that: (a) xm ∗ xn = xm+n . (b) if m > n and xm = xn , then xm−n−1 = x−1 Use (b) to show that if (G, ∗) is a finite group and H is a non-empty subset of G such that x, y ∈ H =⇒ x ∗ y ∈ H, then H is a subgroup of (G, ∗). [Hint: if x ∈ H, show that (at least) two of the elements in list x, x2 , x3 , x4 , . . . must be equal.] 5. Let (G, ∗) be a group and let H ⊆ G. Suppose that ∗H is defined by x ∗H y = x ∗ y for x, y ∈ H, and ∗H is a well-defined operation on H such that (H, ∗H ) is a group. Let e be the identity element of the group (G, ∗). Let eH be the identity element of the group (H, ∗H ). If x ∈ G, let x−1 be the inverse of x with respect to the operation ∗. If x ∈ H, let x−1,H be the inverse of x with respect to the operation ∗H . Prove that: (a) H 6= ∅ (b) x, y ∈ H =⇒ x ∗ y ∈ H (c) e = eH [Hint: show that eH ∗ eH = eH and consider e−1 H .] (d) x ∈ H =⇒ x−1 = x−1,H (e) x ∈ H =⇒ x−1 ∈ H (f) H is a subgroup of (G, ∗) [Together with Theorem 9, this exercise shows that H is a subgroup of (G, ∗) if and only if (H, ∗H ) is a group].