Problem 1TFE: True or False
Label each of the following statements as either true or false.
The set of all... Problem 2TFE: True or False
Label each of the following statements as either true or false.
2. The set of nonzero... Problem 3TFE: Label each of the following statements as either true or false. The identity element in a group G is... Problem 4TFE: True or False
Label each of the following statements as either true or false.
4. If is an abelian... Problem 5TFE Problem 6TFE: True or False
Label each of the following statements as either true or false.
6. The set of all... Problem 7TFE: Label each of the following statements as either true or false. The Cayley table for a group will... Problem 8TFE Problem 9TFE: True or False
Label each of the following statements as either true or false.
9. The nonzero... Problem 10TFE: True or False
Label each of the following statements as either true or false.
10. The nonzero... Problem 11TFE: True or False
Label each of the following statements as either true or false.
11. The invertible... Problem 1E Problem 2E Problem 3E Problem 4E Problem 5E: In Exercises 114, decide whether each of the given sets is a group with respect to the indicated... Problem 6E: Exercises
In Exercises, decide whether each of the given sets is a group with respect to the... Problem 7E: Exercises
In Exercises, decide whether each of the given sets is a group with respect to the... Problem 8E: Exercises
In Exercises, decide whether each of the given sets is a group with respect to the... Problem 9E: Exercises
In Exercises, decide whether each of the given sets is a group with respect to the... Problem 10E: In Exercises 114, decide whether each of the given sets is a group with respect to the indicated... Problem 11E Problem 12E Problem 13E Problem 14E: In Exercises 114, decide whether each of the given sets is a group with respect to the indicated... Problem 15E: In Exercises and, the given table defines an operation of multiplication on the set. In each case,... Problem 16E: In Exercises 15 and 16, the given table defines an operation of multiplication on the set S={... Problem 17E: In Exercises, let the binary operation be defined on by the given rule. Determine in each case... Problem 18E: In Exercises, let the binary operation be defined on by the given rule. Determine in each case... Problem 19E: In Exercises, let the binary operation be defined on by the given rule. Determine in each case... Problem 20E: In Exercises 1724, let the binary operation be defined on by the given rule. Determine in each... Problem 21E: In Exercises 1724, let the binary operation be defined on by the given rule. Determine in each... Problem 22E: In Exercises, let the binary operation be defined on by the given rule. Determine in each case... Problem 23E: In Exercises, let the binary operation be defined on by the given rule. Determine in each case... Problem 24E: In Exercises, let the binary operation be defined on by the given rule. Determine in each case... Problem 25E Problem 26E Problem 27E Problem 28E Problem 29E Problem 30E Problem 31E Problem 32E Problem 33E: a. Let G={ [ a ][ a ][ 0 ] }n. Show that G is a group with respect to multiplication in n if and... Problem 34E: 34. Let be the set of eight elements with identity element and noncommutative multiplication... Problem 35E: 35. A permutation matrix is a matrix that can be obtained from an identity matrix by interchanging... Problem 36E: Consider the matrices R=[ 0110 ] H=[ 1001 ] V=[ 1001 ] D=[ 0110 ] T=[ 0110 ] in GL(2,), and let G={... Problem 37E: Prove or disprove that the set of all diagonal matrices in Mn() forms a group with respect to... Problem 38E: 38. Let be the set of all matrices in that have the form
with all three numbers , , and... Problem 39E: 39. Let be the set of all matrices in that have the form
for arbitrary real numbers , , and .... Problem 40E: 40. Prove or disprove that the set in Exercise is a group with respect to addition.
38. Let be... Problem 41E: 41. Prove or disprove that the set in Exercise is a group with respect to addition.
39. Let be... Problem 42E: 42. For an arbitrary set , the power set was defined in Section by , and addition in was... Problem 43E: Write out the elements of P(A) for the set A={ a,b,c }, and construct an addition table for P(A)... Problem 44E: Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union.... Problem 45E: 45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )
Problem 46E: In Example 3, the group S(A) is nonabelian where A={ 1,2,3 }. Exhibit a set A such that S(A) is... Problem 47E: 47. Find the additive inverse of in the given group.
a. b.
Problem 48E Problem 49E: 49. Find the multiplicative inverse of in the given group.
a. ... Problem 50E: 50. Find the multiplicative inverse of in the given group.
a. ... Problem 51E: Prove that the Cartesian product 24 is an abelian group with respect to the binary operation of... Problem 52E: Let G1 and G2 be groups with respect to addition. Define equality and addition in the Cartesian... format_list_bulleted