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- If a is an element of order m in a group G and ak=e, prove that m divides k.In Exercises 114, decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in Definition 3.1 that fails to hold. The set of all multiples of a positive integer n is group with operation multiplication.Let n be appositive integer, n1. Prove by induction that the set of transpositions (1,2),(1,3),...,(1,n) generates the entire group Sn.
- 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.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.
- The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is. A4=D4.12. 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.Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?
- In Exercises 114, decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in Definition 3.1 that fails to hold. The set of all positive irrational numbers with operation multiplication.Find the order of each of the following elements in the multiplicative group of units . for for for for