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mathematics
a first course in abstract algebra

Questions and Answers of A First Course In Abstract Algebra

Find the number of different partitions of a set having the given number of elements.2 element
Find all orders of subgroups of the given group. Z6
Give a set different from any of those described in the examples of the text and not a set of numbers. Define two different binary operations * and *' on this set. Be sure that your set is well
Mark each of the following true or false._____ a. A group may have more than one identity element._____b. Any two groups of three elements are isomorphic._____c. In a group, each linear equation has
Find the number of different partitions of a set having the given number of elements.3 element
Continuing the ideas of Exercise 24 can a binary structure have a left identity element eL and a right identity element eR where eL ≠ eR? If so, give an example, using an operation on a finite
Find all orders of subgroups of the given group.Z8
Find the number of different partitions of a set having the given number of elements.4 element
Recall that if f : A → B is a one-to-one function mapping A onto B, then f-1 (b) is the unique a ∈ A such that f(a) = b. Prove that if ∅: S-+ S' is an isomorphism of (S, *) with (S', *'),
Compute the given expression using the indicated modular addition. 2√2+√/323√/2
Which of the following groups are cyclic? For each cyclic group, list all the generators of the group.G1 = (Z, +) G2 = (Q, +) G3 = ( Q+, ·) G4 = (6Z, +) G5= {6n |n ∈ Z}
Find all orders of subgroups of the given group.Z12
Either prove the statement or give a counterexample.Every binary operation on a set consisting of a single element in both commutative and associative.
Give at most a two-sentence synopsis of the proof in Theorem 4.16 that an equation ax = b has a unique solution in a group.
Find the number of different partitions of a set having the given number of elements.5 element
Find the order of the cyclic subgroup of the given group generated by the indicated element.The subgroup of Z4 generated by 3
Prove that if ∅ : S → S' is an isomorphism of (S, *) with (S', *') and Ψ : S'→ S11 is an isomorphism of (S', *') with (S", *"), then the composite function 1/J o ¢ is an isomorphism of (S, *)
Find all orders of subgroups of the given group.Z20
Explain why the expression 5 + 6 8 in R6 makes no sense.
Either prove the statement or give a counterexample.Every commutative binary operation on a set having just two elements is associative.
From our intuitive grasp of the notion of isomorphic groups, it should be clear that if ∅ : G ➔ G' is a group isomorphism, then ∅(e) is the identity e' of G'. Recall that Theorem 3.14 gave a
Consider a partition of a set S. The paragraph following Definition 0.18 explained why the relation x R y if and only if x and y are in the same cell satisfies the symmetric condition for an
Find all orders of subgroups of the given group.Z17
Show that if G is a finite group with identity e and with an even number of elements, then there is a ≠ e in G such that a * a = e.
Let F be the set of all real-valued functions having as domain the set R. of all real numbers. Example 2.7 defined the binary operations +, -, •, and o on F. Either prove the given statement
Find all solutions x of the given equation.x +15 7 = 3 in Z15
Determine whether the given relation is an equivalence relation on the set. Describe the partition arising from each equivalence relation.n R m in Z if nm > 0
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.An element a of a group G has order n ∈ Z+
Let R* be the set of all real numbers except 0. Define * on R* by letting a * b = |a |b.a. Show that * gives an associative binary operation on R*. b. Show that there is a left identity for *
Find all solutions x of the given equation. 32 3.77 x + 2x ³ = ³ in R27
Find the order of the cyclic subgroup of the given group generated by the indicated element. The subgroup of Us generated by cos 45+ i sin
Let F be the set of all real-valued functions having as domain the set R. of all real numbers. Example 2.7 defined the binary operations +, -, •, and o on F. Either prove the given statement
Find the order of the cyclic subgroup of the given group generated by the indicated element. The subgroup of Ug generated by cos 3 + i sin 377
Determine whether the given relation is an equivalence relation on the set. Describe the partition arising from each equivalence relation.x R y in R if x ≥ y
Give a careful proof for a skeptic that the indicated property of a binary structure (S, *) is indeed a structural property. (In Theorem 3.14, we did this for the property, "There is an identity
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.The greatest common divisor of two positive
If * is a binary operation on a set S, an element x of S is an idempotent for * if x * x = x. Prove that a group has exactly one idempotent element.
Find the order of the cyclic subgroup of the given group generated by the indicated element. The subgroup of Us generated by cos 5 + i sin
Let F be the set of all real-valued functions having as domain the set R. of all real numbers. Example 2.7 defined the binary operations +, -, •, and o on F. Either prove the given statement
Find all solutions x of the given equation.X +7 X = 3 in Z7
Determine whether the given relation is an equivalence relation on the set. Describe the partition arising from each equivalence relation.x R y in R if |x| = |Y|
Give a careful proof for a skeptic that the indicated property of a binary structure (S, *) is indeed a structural property. (In Theorem 3.14, we did this for the property, "There is an identity
Show that every group G with identity e and such that x * x = e for all x ∈ G is abelian.
Let F be the set of all real-valued functions having as domain the set R. of all real numbers. Example 2.7 defined the binary operations +, -, •, and o on F. Either prove the given statement
Show that if (a * b )2 = a2 * b2 for a and b in a group G, then a * b = b * a.
Find all solutions x of the given equation.X +7 X + 7 X = 5 in Z7
Determine whether the given relation is an equivalence relation on the set. Describe the partition arising from each equivalence relation.x R y in R if |x - y| ≤ 3
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 00 1 01
Either give an example of a group with the property described, or explain why no example exists.A finite group that is not cyclic
Let G be an abelian group and let en = c * c * • • • * c for n factors c, where c ∈ G and n ∈ Z+. Give a mathematical induction proof that (a * b )n = (an ) * (bn ) for all a, b ∈ G.
Let F be the set of all real-valued functions having as domain the set R. of all real numbers. Example 2.7 defined the binary operations +, -, •, and o on F. Either prove the given statement
Find all solutions x of the given equation.X + 12 X = 2 in Z12
Determine whether the given relation is an equivalence relation on the set. Describe the partition arising from each equivalence relation.n R m in Z+ if n and m have the same number of digits in the
There are 16 possible binary structures on the set {a, b} of two elements. How many nonisomorphic (that is, structurally different) structures are there among these 16? Phrased more precisely in
Either give an example of a group with the property described, or explain why no example exists.An infinite group that is not cyclic
Let G be a group with a finite number of elements. Show that for any a ∈ G, there exists an n ∈ Z+ such that an = e. See Exercise 33 for the meaning of an •
Let F be the set of all real-valued functions having as domain the set R. of all real numbers. Example 2.7 defined the binary operations +, -, •, and o on F. Either prove the given statement
Find all solutions x of the given equation.X + 4 X +4 X +4 X = 0 in Z4
Determine whether the given relation is an equivalence relation on the set. Describe the partition arising from each equivalence relation.n R m in Z+ if n and m have the same final digit in the usual
Either give an example of a group with the property described, or explain why no example exists.A cyclic group having only one generator
Let F be the set of all real-valued functions having as domain the set R. of all real numbers. Example 2.7 defined the binary operations +, -, •, and o on F. Either prove the given statement
Using set notation of the form {#,#.#.• ••} for an infinite set, write the residue classes modulo n in Z+ discussed in Example 0.17 for the indicated value of n.a. n = 2 b. n = 3 c. n
Either give an example of a group with the property described, or explain why no example exists.An infinite cyclic group having four generators
Let G be a group and let a, b ∈ G. Show that (a* b)' =a' * b' if and only if a* b = b * a.
There is an isomorphism of U7 with Z7 in which ζ = ei(2π/7) ↔ 4. Find the element in Z7 to which ζm must correspond form = 0, 2, 3, 4, 5, and 6.
Let n ∈ Z+ and let ~ be defined on Z by r ~ s if and only if r - s is divisible by n, that is, if and only if r - s = nq for some q ∈ Z.a. Show that ~ is an equivalence relation on Z. (It
Students often misunderstand the concept of a one-to-one function (mapping). I think I know the reason. You see, a mapping ∅ : A → B has a direction associated with it, from A to B. It seems
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 subgroup of a group G is a subset H of G
Either give an example of a group with the property described, or explain why no example exists.A finite cyclic group having four generators
Let G be a group and suppose that a * b * c = e for a, b, c ∈ G. Show that b * c * a= e also.
Suppose that * is an associative and commutative binary operation on a set S. Show that H = {a ∈ S| a* a = a} is closed under *· (The elements of H are idempotents of the binary operation *·)
Why can there be no isomorphism of U6 with Z6 in which ζ = ei(π/3) corresponds to 4?
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 group G is cyclic if and only if there
Derive the formulassin( a + b) = sin a cos b + cos a sin bandcos(a + b) = cos a cos b - sin a sin b
Let G be a cyclic group with generator a, and let G' be a group isomorphic to G. If ∅ : G → G' is an isomorphism, show that, for every x ∈ G, ∅(x) is completely determined by the value
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 = 12
Show by means of all example that it is possible for the quadratic equation x2 = e to have more than two solutions in some group G with identity e.
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 = 8
Prove that a nonempty set G, together with an associative binary operation * on G such that a * x = b and y * a = b have solutions in G for all a, b ∈ G, is a group.
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 = 6
Prove that a set G, together with a binary operation * on G satisfying the left axioms 1, 2, and 3 is a group.
Let ( G, •) be a group. Consider the binary operation * on the set G defined byfor a, b ∈ G. Show that ( G, *) is a group and that ( G, *) is actually isomorphic to ( G, •). a*b = b.a D
a. Derive a formula for cos 3θ in terms of sinθand cosθ using Euler's formula.b. Derive the formula cos 3θ = 4 cos3 θ - 3 cosθ from part (a) and the identity sin2 θ + cos2θ= 1.
Let G be a group and let g be one fixed element of G. Show that the map ig, such that ig (x) = gxg' for x ∈ G, is an isomorphism of G with itself.
Recall the power series expansionsfrom calculus. Derive Euler's formula eiθ = cosθ + i sinθ formally from these three series expansions. e* = 1+x+ sin x = x cos x = 1 - - x² + + + 2! 3! 4! x
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 G is cyclic, then G' is cyclic.
Show that if H and K are subgroups of an abelian group G, then {hk | h ∈ H and k ∈ K} is a subgroup of G.
Let r ands be positive integers. Show that {nr + ms | n, m ∈ Z} is a subgroup of Z.
Show that a nonempty subset H of a group G is a subgroup of G if and only if ab-1 ∈ H for all a, b ∈ H.
Let a and b be elements of a group G. Show that if ab has finite order n, then ba also has order n.
Prove that a cyclic group with only one generator can have at most 2 elements.
Let r and s be positive integers. a. Define the least common multiple of r and s as a generator of a certain cyclic group. b. Under what condition is the least common multiple of r ands
Prove that if G is an abelian group, written multiplicatively, with identity element e, then all elements x of G satisfying the equation x2 = e form a subgroup H of G.
Show that a group that has only a finite number of subgroups must be a finite group.
Show that if a ∈ G, where G is a finite group with identity e, then there exists n ∈ Z+ such that an = e.
Show by a counterexample that the following "converse" of Theorem 6.6 is not a theorem: "If a group G is such that every proper subgroup is cyclic, then G is cyclic."
Let a nonempty finite subset H of a group G be closed under the binary operation of G. Show that H is a subgroup of G.
Let G be a group and suppose a ∈ G generates a cyclic subgroup of order 2 and is the unique such element. Show that ax = xa for all x ∈ G.
Let p and q be distinct prime numbers. Find the number of generators of the cyclic group Zpq.
Let G be a group and let a be one fixed element of G. Show that Ha = {x ∈ G | xa = ax} is a subgroup of G.

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