Let G be a group and a,b∈ G such that ab = ba. Let o(a)= m and o(b) = n. Show that there exists an element c∈G such that o(c) is the least common multiple of m and n.
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Let G be a group and a,b∈ G such that ab = ba. Let o(a)= m and o(b) = n. Show that there exists an element c∈G such that o(c) is the least common multiple of m and n.
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- Let a,b,c, and d be elements of a group G. Find an expression for (abcd)1 in terms of a1,b1,c1, and d1.If a is an element of order m in a group G and ak=e, prove that m divides k.38. Let be the set of all matrices in that have the form with all three numbers , , and nonzero. Prove or disprove that is a group with respect to multiplication.
- 12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.In Exercises 15 and 16, the given table defines an operation of multiplication on the set S={ e,a,b,c }. In each case, find a condition in Definition 3.1 that fails to hold, and thereby show that S is not a group. See Figure 3.7 e a b c e e a b c a e a b c b e a b c c e a b cIn Exercises and, the given table defines an operation of multiplication on the set. In each case, find a condition in Definition that fails to hold, and thereby show that is not a group. 15. See Figure.
- 42. For an arbitrary set , the power set was defined in Section by , and addition in was defined by Prove that is a group with respect to this operation of addition. If has distinct elements, state the order of .39. Let be the set of all matrices in that have the form for arbitrary real numbers , , and . Prove or disprove that is a group with respect to multiplication.Label each of the following statements as either true or false. The Generalized Associative Law applies to any group, no matter what the group operation is.