Determine whether the following is a semigroup, monoid, or a group. 1. G=Z, a*b = a + ab 2. G=2Z, a*b=a+b 7
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- 9. Find all elements in each of the following groups such that . under addition. under multiplication.Label each of the following statements as either true or false. The Generalized Associative Law applies to any group, no matter what the group operation is.(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.
- 6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.Exercises 8. Let be an element of order in a group. Find the order of each of the following. a. b. c. d. e. f. g. h.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 .