10. Show if the given set under the given binary operation is a group. If it is a group, show if it is abelian and determine if it is cyclic. Let be defined on 3 = {3n| n M} by a W b = a+ b. - end -
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- If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.If a is an element of order m in a group G and ak=e, prove that m divides k.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 .
- 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.Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.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.
- 27. Suppose that is a nonempty set that is closed under an associative binary operation and that the following two conditions hold: There exists a left identity in such that for all . Each has a left inverse in such that . Prove that is a group by showing that is in fact a two-sided identity for and that is a two-sided inverse of .Let H1 and H2 be cyclic subgroups of the abelian group G, where H1H2=0. Prove that H1H2 is cyclic if and only if H1 and H2 are relatively prime.42. For an arbitrary set , the power set was defined in Section by , and addition in was defined by Prove that is a group with respect to this operation of addition. If has distinct elements, state the order of .
- Exercises 18. Suppose and let be defined by . Prove or disprove that is an automorphism of the additive group .(See Exercise 31.) Suppose G is a group that is transitive on 1,2,...,n, and let ki be the subgroup that leaves each of the elements 1,2,...,i fixed: Ki=gGg(k)=kfork=1,2,...,i For i=1,2,...,n. Prove that G=Sn if and only if HiHj for all pairs i,j such that ij and in1. A subgroup H of the group Sn is called transitive on B=1,2,....,n if for each pair i,j of elements of B there exists an element hH such that h(i)=j. Suppose G is a group that is transitive on 1,2,....,n, and let Hi be the subgroup of G that leaves i fixed: Hi=gGg(i)=i For i=1,2,...,n. Prove that G=nHi.Prove that if r and s are relatively prime positive integers, then any cyclic group of order rs is the direct sum of a cyclic group of order r and a cyclic group of order s.