Let X, Y be cyclic groups of order 2, 3 respectively. Find the orders of the elements of XxY.
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- 45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )Find the right regular representation of G as defined Exercise 11 for each of the following groups. a. G={ 1,i,1,i } from Example 1. b. The octic group D4={ e,,2,3,,,, }.Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)
- Exercises 9. For each of the following values of, find all distinct generators of the cyclic group under addition. a. b. c. d. e. f.If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.
- 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=1927. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.Find all subgroups of the octic group D4.