A. Show that each of the following is not a group. (Hint: identify and show the group property that is not satisfied). 1) * defined on Z by a * b = la + b| 2) • defined on Q by a • b = * 3) * defined on R by a i a+b b a+b- ab
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- If a is an element of order m in a group G and ak=e, prove that m divides k.(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.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 .
- 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.In Exercises and, the given table defines an operation of multiplication on the set. In each case, find a condition in Definition that fails to hold, and thereby show that is not a group. 15. See Figure.In Exercises 15 and 16, the given table defines an operation of multiplication on the set S={ e,a,b,c }. In each case, find a condition in Definition 3.1 that fails to hold, and thereby show that S is not a group. See Figure 3.7 e a b c e e a b c a e a b c b e a b c c e a b c
- Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.Label each of the following statements as either true or false. Two groups can be isomorphic even though their group operations are different.
- True or False Label each of the following statements as either true or false. A group may have more than one identity element.Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.In Exercises 3 and 4, let be the octic group in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let be the subgroup of the octic group . Find the distinct left cosets of in , write out their elements, partition into left cosets of , and give . Find the distinct right cosets of in , write out their elements, and partition into right cosets of . Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group of rigid motions of a square The elements of the group are as follows: 1. the identity mapping 2. the counterclockwise rotation through about the center 3. the counterclockwise rotation through about the center 4. the counterclockwise rotation through about the center 5. the reflection about the horizontal line 6. the reflection about the diagonal 7. the reflection about the vertical line 8. the reflection about the diagonal . The dihedral group of rigid motions of the square is also known as the octic group. The multiplication table for is requested in Exercise 20 of this section.