QUESTION 5 Show that ifevery element in a group G is equal to its own inverse, then G is Abelian.
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- Prove part c of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)2=a2b2.
- 1.Prove part of Theorem . Theorem 3.4: Properties of Group Elements Let be a group with respect to a binary operation that is written as multiplication. The identity element in is unique. For each, the inverse in is unique. For each . Reverse order law: For any and in ,. Cancellation laws: If and are in , then either of the equations or implies that .True or False Label each of the following statements as either true or false. Let H1,H2 be finite groups of an abelian group G. Then | H1H2 |=| H1 |+| H2 |.In Example 3, the group S(A) is nonabelian where A={ 1,2,3 }. Exhibit a set A such that S(A) is abelian. Example 3. We shall take A={ 1,2,3 } and obtain an explicit example of S(A). In order to define an element f of S(A), we need to specify f(1), f(2), and f(3). There are three possible choices for f(1). Since f is to be bijective, there are two choices for f(2) after f(1) has been designated, and then only once choice for f(3). Hence there are 3!=321 different mappings f in S(A).
- 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.11. Show that is a generating set for the additive abelian group if and only ifIn Exercises 3 and 4, let G be the octic group D4=e,,2,3,,,, in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group D4 of rigid motions of a square The elements of the group D4 are as follows: 1. the identity mapping e=(1) 2. the counterclockwise rotation =(1,2,3,4) through 900 about the center O 3. the counterclockwise rotation 2=(1,3)(2,4) through 1800 about the center O 4. the counterclockwise rotation 3=(1,4,3,2) through 2700 about the center O 5. the reflection =(1,4)(2,3) about the horizontal line h 6. the reflection =(2,4) about the diagonal d1 7. the reflection =(1,2)(3,4) about the vertical line v 8. the reflection =(1,3) about the diagonal d2. The dihedral group D4=e,,2,3,,,, of rigid motions of the square is also known as the octic group. The multiplication table for D4 is requested in Exercise 20 of this section.
- Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.40. Find the commutator subgroup of each of the following groups. a. The quaternion group . b. The symmetric group .Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.