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mathematics
a first course in abstract algebra

Questions and Answers of A First Course In Abstract Algebra

In Exercises, decide whether the object described is indeed a set (is well defined). Give an alternate description of each set.{x ∈ Q| x may be written with denominator greater than 100}
Compute the given arithmetic expression and give the answer in the form a + bi for a, b ∈ R.(1 - i)5 (Use the binomial theorem.)
Give the table for the group having the indicated digraph. In each digraph, take е as identity element. List the identity e first in your table, and list the remaining elements alphabetically, so
Give the table for the group having the indicated digraph. In each digraph, take e as identity element. List the identity e first in your table, and list the remaining elements alphabetically, so
Determine whether the given map ∅ is an isomorphism of the first binary structure with the second. If it is not an isomorphism, why not?(M1(R), ·) with (R, ·) where ∅(A) is the determinant of
Show that the group (U, •) is not isomorphic to either (R +) or (R*,•). (All three groups have cardinality |R|.)
Determine whether the binary operation * defined is commutative and whether * is associative.* defined on (Q by letting a * b = ab /2
Find the number of generators of a cyclic group having the given order.8
List the elements of the subgroup generated by the given subset.The subset {4, 6} of Z12
Find the quotient and remainder, according to the division algorithm, when n is divided by m.n = 42, m = 9
Determine whether the given subset of the complex numbers is a subgroup of the group CC of complex numbers under addition.R
Determine whether the binary operation * gives a group structure on the given set. If no group results, give the first axiom in the order G1, G2, G3 from Definition 4.1 that does not hold.Let* be
List the elements of the subgroup generated by the given subset.The subset {2, 3} of Z12
Determine whether the binary operation * gives a group structure on the given set. If no group results, give the first axiom in the order G1, G2, G3 from Definition 4.1 that does not hold.Let* be
Determine whether the given subset of the complex numbers is a subgroup of the group CC of complex numbers under addition.Q+
Find the quotient and remainder, according to the division algorithm, when n is divided by m.n = -42, m = 9
List the elements of the subgroup generated by the given subset.The subset {8, 10} of Z18
Determine whether the given map ∅ is an isomorphism of the first binary structure with the second. If it is not an isomorphism, why not?(Q, +) with (Q, +) where ∅(x) = x/2 for x ∈ Q
Find the quotient and remainder, according to the division algorithm, when n is divided by m.n = 50,m = 8
Determine whether the given subset of the complex numbers is a subgroup of the group CC of complex numbers under addition.The set πQ of rational multiples of π
Find the greatest common divisor of the two integers. 32 and 24
Give an example of an abelian group G where G has exactly 1000 elements.
Determine whether the binary operation * defined is commutative and whether * is associative.* defined on (Q by letting a * b = ab + 1
Find the number of generators of a cyclic group having the given order.5
Determine whether the given set of invertible n x n matrices with real number entries is a subgroup of GL(n, R).The diagonal n x n matrices with no zeros on the diagonal
Find |3 - 4i|.
In Exercises, decide whether the object described is indeed a set (is well defined). Give an alternate description of each set.{x ∈ Q| x may be written with positive denominator less than 4}
Repeat Exercise 47 for the general situation of the set H of all solutions x of the equation xn = e for a fixed integer n ≥ 1 in an abelian group G with identity e.Data from exercise 47Prove that
Is it obvious from a Cayley digraph of a group whether or not the group is cyclic?
Determine whether the given set of matrices under the specified operation, matrix addition or multiplication, is a group. Recall that a diagonal matrix is a square matrix whose only nonzero entries
Let F be the set of all functions f mapping R. into R. that have derivatives of all orders. Follow the instructions for Exercises 2 through 10.(F, +) with (F, +) where ∅(f)(x) = f0x f (t) dt
Determine whether the given set of invertible n x n matrices with real number entries is a subgroup of GL(n, R).The set of all n x n matrices A such that (AT)A = ln • [These matrices are called
An isomorphism of a group with itself is an automorphism of the group. Find the number of automorphisms of the given group.Z6
How many different commutative binary operations can be defined on a set of 2 elements? on a set of 3 elements? on a set of n elements?
Recall that for a, b ∈ R and a < b, the closed interval [a, b] in R is defined by [a, b] = {x ∈ R I a ≤ x ≤ b}. Show that the given intervals have the same cardinality by giving a formula
Write the given complex number z in the polar form |z|(p + qi) where |p +qi|= 1.12 + 5i
The large outside triangle in Fig. 7.9(b) exhibits the cyclic subgroup {O, 2, 4} of Z6 • Does the smaller inside triangle similarly exhibit a cyclic subgroup of Z6 ? Why or why not?
Determine whether the given set of matrices under the specified operation, matrix addition or multiplication, is a group. Recall that a diagonal matrix is a square matrix whose only nonzero entries
Let F be the set of all functions f mapping R. into R. that have derivatives of all orders. Follow the instructions for Exercises 2 through 10. (F. +) with (F, +) where p(f)(x) = [fő f(t)dt] dx
Let F be the set of all real-valued functions with domain R and let F̅ be the subset of F consisting of those functions that have a nonzero value at every point in R. Determine whether the given
An isomorphism of a group with itself is an automorphism of the group. Find the number of automorphisms of the given group.Z8
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication.A binary operation* is commutative if and only
Write the given complex number z in the polar form |z|(p + qi) where |p +qi|= 1.-3 + 5i
Show that S = {x ∈ R I 0 < x < 1} has the same cardinality as R For any set A, we denote by P(A) the collection of all subsets of A. For example, if A = {a, b, c, d}, then {a, b, d} ∈
The generating set S = {1, 2} for Z6 contains more generators than necessary, since 1 is a generator for the group. Nevertheless, we can draw a Cayley digraph for Z6 with this generating set S.
Determine whether the given set of matrices under the specified operation, matrix addition or multiplication, is a group. Recall that a diagonal matrix is a square matrix whose only nonzero entries
Let F be the set of all functions f mapping R. into R. that have derivatives of all orders. Follow the instructions for Exercises 2 through 10.(F, •) with (F, •) where ∅(f)(x) = x · f(x)
Let F be the set of all real-valued functions with domain R and let F̅ be the subset of F consisting of those functions that have a nonzero value at every point in R. Determine whether the given
An isomorphism of a group with itself is an automorphism of the group. Find the number of automorphisms of the given group.Z
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication.A binary operation * on a set S is associative
List the elements of the power set of the given set and give the cardinality of the power set.a. øb. {a}c. {a, b} d. {a, b, c}For any set A, we denote by P(A) the collection of all subsets of
Draw a Cayley digraph for Z8 taking as generating set S = {2, 5}.
Determine whether the given set of matrices under the specified operation, matrix addition or multiplication, is a group. Recall that a diagonal matrix is a square matrix whose only nonzero entries
let ∅: G → G' be an isomorphism of a group ( G, *) with a group ( G', *'). Write out a proof to convince a skeptic of the intuitively clear statement.If H is a subgroup of G, then ∅[HJ =
Find the order of the cyclic subgroup of the given group generated by the indicated element.The subgroup of the multiplicative group G of invertible 4 x 4 matrices generated by 00 1 00 0010 100 0 0
Find the order of the cyclic subgroup of the given group generated by the indicated element.The subgroup of the multiplicative group G of invertible 4 x 4 matrices generated by 0 10 0 000
Example 1.15 asserts that there is an isomorphism of U8 with Z8 in which ζ= ei(π/4) ↔ 5 and ζ2 ↔2. Find the element of Z8 that corresponds to each of the remaining six elements ζm in U8 form=
Suppose that * is an associative binary operation on a set S. Let H = {a ∈ S| a * x = x * a for all x ∈ S}. Show that H is closed under *· (We think of H as consisting of all elements of S that
The generators of the cyclic multiplicative group Un of all nth roots of unity in C are the primitive nth roots of unity. Find the primitive nth roots of unity for the given value of n.n = 4
Let A = {l, 2} and let B = {3, 4, 5}.a. Illustrate, using A and B, why we consider that 2 + 3 = 5. Use similar reasoning with sets of your own choice to decide what you would consider to be the value
Prove that if * is an associative and commutative binary operation on a set S, thenfor all a, b, c, d ∈ S. Assume the associative law only for triples as in the definition, that is, assume onlyfor
Let F be the set of all real-valued functions with domain R and let F̅ be the subset of F consisting of those functions that have a nonzero value at every point in R. Determine whether the given
Find the number of elements in the indicated cyclic group.The cyclic subgroup of Z30 generated by 25
Determine whether the definition of * does give a binary operation on the set. In the event that * is not a binary operation, state whether Condition 1, Condition 2, or both of these conditions on
For any set A, finite or infinite, let BA be the set of all functions mapping A into the set B = {0, 1 }. Show that the cardinality of BA is the same as the cardinality of the set P(A). Then try to
Determine whether the given set of matrices under the specified operation, matrix addition or multiplication, is a group. Recall that a diagonal matrix is a square matrix whose only nonzero entries
Draw digraphs of the two possible structurally different groups of order 4, taking as small a generating set as possible in each case. You need not label vertices.
The map ∅ (Q → (Q defined by ∅(x) = 3x - 1 for x ∈ Q is one to one and onto Q. Give the definition of a binary operation * on Q such that ∅ is an isomorphism mappinga. (Q, +) onto (Q,
Let F be the set of all real-valued functions with domain R and let F̅ be the subset of F consisting of those functions that have a nonzero value at every point in R. Determine whether the given
Find the number of elements in the indicated cyclic group.The cyclic subgroup of Z42 generated by 30
Determine whether the definition of * does give a binary operation on the set. In the event that * is not a binary operation, state whether Condition 1, Condition 2, or both of these conditions on
Determine whether the definition of * does give a binary operation on the set. In the event that * is not a binary operation, state whether Condition 1, Condition 2, or both of these conditions on
Let S be the set of all real numbers except -1. Define * on S by a * b = a + b + ab.a. Show that * gives a binary operation on S.b. Show that (S, *) is a group. c. Find the solution of the
∅∅The map ∅: Q → Q defined by ∅(x) = 3x - 1 for x ∈ Q is one to one and onto Q. Give the definition of a binary operation * on Q such that ∅ is an isomorphism mappinga. (Q, •) onto
Let F be the set of all real-valued functions with domain R and let F̅ be the subset of F consisting of those functions that have a nonzero value at every point in R. Determine whether the given
Determine whether the definition of * does give a binary operation on the set. In the event that * is not a binary operation, state whether Condition 1, Condition 2, or both of these conditions on
Determine whether the definition of * does give a binary operation on the set. In the event that * is not a binary operation, state whether Condition 1, Condition 2, or both of these conditions on
This exercise shows that there are two nonisomorphic group structures on a set of 4 elements.Let the set be { e, a, b, c}, with e the identity element for the group operation. A group table would
Nine groups are given below. Give a complete list of all subgroup relations, of the form Gi ≤ GJ, that exist between these given groups G1, G2, • • •, G9• G1 = Z under
Find the number of elements in the indicated cyclic group.The cyclic subgroup of the group C* of Exercise 19 generated by (1 + i) √2Data from ex. 19The cyclic subgroup ( i ) of the group C* of
Determine whether the definition of * does give a binary operation on the set. In the event that * is not a binary operation, state whether Condition 1, Condition 2, or both of these conditions on
Consider our axioms G1, G2, and G3 for a group. We gave them in the order G1, G2, and G3. Conceivable other orders to state the axioms are G1G3 G2, G2 G1G3, G2 G3 G1. G3 G1 G2, and G3
How many numbers in the interval 0 ≤ x ≤ 1 can be expressed in the form .##, where each # is a digit 0, 1, 2, 3, · · •, 9? How many are there of the form.#####? Following this idea, and
Continuing the idea in the preceding exercise and using Exercises 18 and 19, use exponential notation to fill in the three blanks to give a list of five cardinal numbers, each of which is greater
According to Exercise, there are 16 possible binary operations on a set of 2 elements. How many of these give a structure of a group? How many of the 19,683 possible binary operations on a set of 3
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication.A function∅: S-+ S' is an isomorphism if and
Describe all the elements in the cyclic subgroup of GL(2, R) generated by the given 2 x 2 matrix. [5] 0
Find the number of elements in the indicated cyclic group.The cyclic subgroup of the group C* of Exercise 19 generated by 1 + iData from ex. 19The cyclic subgroup ( i ) of the group C* of nonzero
Describe all the elements in the cyclic subgroup of GL(2, R) generated by the given 2 x 2 matrix. 1
Compute the given expression using the indicated modular addition.10 + 17 16
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication.Let * be a binary operation on a set S. An
Find all subgroups of the given group, and draw the subgroup diagram for the subgroups.Z36
Compute the given expression using the indicated modular addition.8 +10 6
Find the number of different partitions of a set having the given number of elements.1 element
Mark each of the following true or false.___ a. If * is any binary operation on any set S, then a * a = a for all a ∈ S.___ b. If * is any commutative binary operation on any set S, then a * (b *
Describe all the elements in the cyclic subgroup of GL(2, R) generated by the given 2 x 2 matrix. [3 0 2
Give a table for a binary operation on the set {e, a, b} of three elements satisfying axioms G2 and G3 for a group but not axiom G1.
Describe all the elements in the cyclic subgroup of GL(2, R) generated by the given 2 x 2 matrix. 0 -2 -27 0
Find all subgroups of the given group, and draw the subgroup diagram for the subgroups.Z8
Compute the given expression using the indicated modular addition.20.5 + 25 19.3

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