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A: Thanks for the question :)And your upvote will be really appreciable ;)
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- 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.Exercises 11. According to Exercise of section, if is prime, the nonzero elements of form a group with respect to multiplication. For each of the following values of , show that this group is cyclic. (Sec. ) a. b. c. d. e. f. 33. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.Use mathematical induction to prove that if a is an element of a group G, then (a1)n=(an)1 for every positive integer n.
- If a is an element of order m in a group G and ak=e, prove that m divides k.Exercises In Exercises, 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 that fails to hold. 8. For a fixed positive integer, the set of all complex numbers such that (that is, the set of all roots of), with operation multiplication.Exercises In Exercises, 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 that fails to hold. 6. The set of all positive rational numbers with operation multiplication.
- Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.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.15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .