1. 3 in (Z90, +90) 2. 4 in U(21), the group of units modulo 21.
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A: Solution:
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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.9. Find all elements in each of the following groups such that . under addition. under multiplication.If a is an element of order m in a group G and ak=e, prove that m divides k.
- Find the order of each of the following elements in the multiplicative group of units . for for for for9. Find all homomorphic images of the octic group.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 be as described in the proof of Theorem. Give a specific example of a positive element of .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 .Prove or disprove that the set of all diagonal matrices in Mn() forms a group with respect to addition.
- The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is. A4=D4.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=196. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.