Section 1.3. Groups 27 1.28. (i) Compute the order, inverse, and parity of α = (1 2)(4 3)(1 3 5 4 2)(1 5)(1 3)(2 3). (ii) What are the respective orders of the permutations in Exercises 1.10 and 1.18 on page 15? 1.29. (i) How many elements of order 2 are there in S5 and in S6? (ii) Make a table for S6 (as the Table 3 on page 24). (iii) How many elements of order 2 are there in Sn? Hint. You may express your answer as a sum. ∗ 1.30. If G is a group, prove that the only element g ∈ G with g2 = g is 1. ∗ 1.31. This exercise gives a shorter list of axioms defining a group. Let H be a semigroup containing an element e such that e ∗ x = x for all x ∈ H and, for every x ∈ H, there is x ∈ H with x ∗ x = e. (i) Prove that if h ∈ H satisfies h ∗ h = h, then h = e. Hint. If h ∗ h = e, evaluate h ∗ h ∗ h in two ways. (ii) For all x ∈ H, prove that x ∗ x = e. Hint. Consider (x ∗ x )2. (iii) For all x ∈ H, prove that x ∗ e = x. Hint. Evaluate x ∗ x ∗ x in two ways. (iv) Prove that if e ∈ H satisfies e ∗ x = x for all x ∈ H, then e = e. Hint. Show that (e )2 = e . (v) Let x ∈ H. Prove that if x ∈ H satisfies x ∗ x = e, then x = x . Hint. Evaluate x ∗ x ∗ x in two ways. (vi) Prove that H is a group. ∗ 1.32. Let y be a group element of order n if n = mt for some divisor m, prove that yt has order m. Hint. Clearly, (yt)m = 1. Use Proposition 1.26 to show that no smaller power of yt is equal to 1. ∗ 1.33. Let G be a group and let a ∈ G have order k. If p is a prime divisor of k and there is x ∈ G with xp = a, prove that x has order pk. ∗ 1.34. Let G = GL(2, Q), let A = [ 0 −1 1 0 ], and let B = 0 1 −1 1 . Show that A4 = I = B6, but that (AB)n = I for all n 0, where I = [ 1 0 0 1 ]. Conclude that AB can have infinite order even though both factors A and B have finite order (of course, this cannot happen in a finite group). ∗ 1.35. If G is a group in which x2 = 1 for every x ∈ G, prove that G must be abelian. [The Boolean groups B(X) in Example 1.18 are such groups.] ∗ 1.36. If G is a group of even order, prove that the number of elements in G of order 2 is odd. In particular, G must contain an element of order 2. Hint. Pair each element with its inverse. 1.37. Let L(n) denote the largest order of an element in Sn. Find L(n) for n = 1, 2,... , 10.

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