Ex. 54 Show that the following additive group is cyclic and give its generator. 1. H2 the set of all even numbers. 2. H1 = {30n + 45m | n, m E Z}.
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- 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=19Exercises 9. For each of the following values of, find all distinct generators of the cyclic group under addition. a. b. c. d. e. f.Exercises 10. For each of the following values of, find all subgroups of the cyclic group under addition and state their order. a. b. c. d. e. f.
- The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is. A4=D4.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.6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.
- If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.Let H1={ [ 0 ],[ 6 ] } and H2={ [ 0 ],[ 3 ],[ 6 ],[ 9 ] } be subgroups of the abelian group 12 under addition. Find H1+H2 and determine if the sum is direct.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={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .