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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.If a is an element of order m in a group G and ak=e, prove that m divides k.9. Find all elements in each of the following groups such that . under addition. under multiplication.
- Find the order of each of the following elements in the multiplicative group of units . for for for forUse mathematical induction to prove that if a is an element of a group G, then (a1)n=(an)1 for every positive integer n.For each of the following values of n, find all distinct generators of the group Un described in Exercise 11. a. n=7 b. n=5 c. n=11 d. n=13 e. n=17 f. n=19
- Exercises 13. For each of the following values of, find all subgroups of the group described in Exercise, addition and state their order. a. b. c. d. e. f.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 .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.