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mathematics
a first course in abstract algebra

Questions and Answers of A First Course In Abstract Algebra

Compute ip1 [H] for the subgroup H = {p0 , µ,1} of the group S3 of Example 8.7.Data from in Example 8.7.An interesting example for us is the group S3 of 3! = 6 elements. Let the set A be { 1,
Describe all subgroups of order ≤ 4 of Z4 x Z4, and in each case classify the factor group of Z4 x Z4 modulo the subgroup by Theorem 11.12. That is, describe the subgroup and say that the
Show how all possible G-sets, up to isomorphism (see Exercise 9), can be formed from the group G.Let Xi for i ∈ I be G-sets for the same group G, and suppose the sets Xi are not necessarily
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 center of a group G contains all
Show how all possible G-sets, up to isomorphism (see Exercise 9), can be formed from the group G.The preceding exercises show that every G-set X is isomorphic to a disjoint union of left coset
Up to isomorphism, how many transitive Z4 sets X are there? Give an example of each isomorphism type, listing an action table of each as in Table 16.10. Take lowercase names a, b, c, and so on for
Compute the indicated quantities for the given homomorphism¢. (See Exercise 46.)Ker(∅) and ∅(25) for ∅ : Z → Z7 such that ∅(1) = 4Data from Exercise 46Let a group G be generated by {
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 normal subgroup Hof G is one
Find both the center and the commutator subgroup of S3 x D4 .
The commutator subgroup of a group G is {a-1b-1 ab | a, b ∈ G}.
Compute the indicated quantities for the given homomorphism¢.Ker(∅) and ∅(18) for ∅ : Z → Z10 such that ∅(1) = 6Data from Exercise 46Let a group G be generated by { ai | i ∈ I},
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 normal subgroup H of G is one satisfying g-1
Mark each of the following true or false. ___ a. Every factor group of a cyclic group is cyclic. ___ b. A factor group of a noncyclic group is again noncyclic. ___ c. R/Z under
Describe all subgroups of order ≤ 4 of Z4 x Z4, and in each case classify the factor group of Z4 x Z4 modulo the subgroup by Theorem 11.12. That is, describe the subgroup and say that the
Compute ip1 [H] for the subgroup H = {p0 , µ,1} of the group S3 of Example 8.7.Data from in Example 8.7.An interesting example for us is the group S3 of 3! = 6 elements. Let the set A be { 1,
Compute the indicated quantities for the given homomorphism¢.Ker (∅) for ∅ : S3 →Z2 in Example 13.3Data from Example 13.3Let Sn be the symmetric group on n letters, and let ∅: Sn → Z2
Give the order of the element in the factor group.(2, 0) + ((4, 4)) in (Z6 x Z8) / ((4, 4))
Compute the indicated quantities for the given homomorphism¢. (See Exercise 46.)Ker(∅) and ∅(18) for ∅ : Z → Z10 such that ∅(1) = 6Data from Exercise 46Let a group G be generated by
Show that if a finite group G contains a nontrivial subgroup of index 2 in G, then G is not simple.
Repeat Exercise 18 for the group Z6.Data from Exercise 18Up to isomorphism, how many transitive Z4 sets X are there? (Use the preceding exercises.) Give an example of each isomorphism type, listing
Compute the indicated quantities for the given homomorphism¢. (See Exercise 46.)Ker(∅) and∅(20) for∅: Z → S8 such that ∅(1) = (1, 4,2, 6)(2, 5, 7)Data from Exercise 46Let a group G
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 automorphism of a group G is a homomorphism
Repeat Exercise 18 for the group S3. List the elements of S3 in the order ι, (1, 2, 3), (1, 3, 2), (2, 3), (1, 3), (1, 2).Data from Exercise 18Up to isomorphism, how many transitive Z4 sets X are
Let F be the additive group of all functions mapping R into R, and let F* be the multiplicative group of all elements of F that do not assume the value 0 at any point of R.Let K be the subgroup of F
Compute the indicated quantities for the given homomorphism¢. (See Exercise 46.)Ker(∅) and ∅(3) for∅ : Z10 → Z20 such that ∅(1) = 8Data from Exercise 46Let a group G be
Let F be the additive group of all functions mapping R into R, and let F* be the multiplicative group of all elements of F that do not assume the value 0 at any point of R.Let K* be the subgroup of
What is the importance of a normal subgroup of a group G?
Compute the indicated quantities for the given homomorphism¢. (See Exercise 46.)Ker(∅) and ∅(14) for∅ : Z24 → S8 where ∅(1) = (2, 5)(1, 4, 6, 7)Data from Exercise 46Let a group G be
A student is asked to show that if H is a normal subgroup of an abelian group G, then G / H is abelian. The student's proof starts as follows:We must show that G / H is abelian. Let a and b be two
Let F be the additive group of all functions mapping R into R, and let F* be the multiplicative group of all elements of F that do not assume the value 0 at any point of R.Let K be the subgroup of
Compute the indicated quantities for the given homomorphism¢. (See Exercise 46.)Ker(∅) and ∅(-3, 2) for ∅: Z x Z → Z where ∅(1, 0) = 3 and ∅ (0, 1) = -5Data from Exercise 46Let a
A torsion group is a group all of whose elements have finite order. A group is torsion free if the identity is the only element of finite order. A student is asked to prove that if G is a torsion
Mark each of the following true or false. ____ a. It makes sense to speak of the factor group G / N if and only if N is a normal subgroup of the group G. ____ b. Every subgroup of an
Let F be the additive group of all functions mapping R into R, and let F* be the multiplicative group of all elements of F that do not assume the value 0 at any point of R.Let K* be the subgroup of
Compute the indicated quantities for the given homomorphism¢. Ker(∅) and ∅(4, 6) for∅ : Z x Z → Z x Z where ∅(1, 0) = (2, -3) and ∅(0, 1) = (-1, 5)Data from Exercise 46Let a
Show that An is a normal subgroup of Sn and compute Sn/ An; that is, find a known group to which Sn/ An is isomorphic.
Let U be the multiplicative group {z ∈ C | |z| = 1}.Let z0 ∈ U. Show that z0 U = {z0z | z ∈ U} is a subgroup of U, and compute U /z0 U.
Compute the indicated quantities for the given homomorphism¢. Ker(∅) and ∅(3, 10) for ∅: Z x Z → S10 where ∅(1, 0) = (3, 5)(2, 4) and ∅(0, 1) = (1, 7)(6, 10, 8, 9)Data from
Complete the proof of Theorem 14.4 by showing that if H is a subgroup of a group G and if left coset multiplication (aH)(bH) = (ab)H is well defined, then Ha ⊆ aH.Data from Theorem 14.4Let H be a
Let U be the multiplicative group {z ∈ C | |z| = 1}.To what group we have mentioned in the text is U/(-1) isomorphic?
How many homomorphisms are there of Z onto Z?
Prove that the torsion subgroup T of an abelian group G is a normal subgroup of G, and that G / T is torsion free. (See Exercise 22.) Data from Exercise 22A torsion group is a group all of whose
Referring to Exercise 27, find all subgroups of S3 (Example 8.7) that are conjugate to {P0, µ2}.Data from Exercise 27A subgroup H is conjugate to a subgroup K of a group G if there exists an inner
Let U be the multiplicative group {z ∈ C | |z| = 1}.Let (ζn= cos(2π/n) + i sin(2π/n) where n ∈ Z+. To what group we have mentioned is U /(ζn) isomorphic?
How many homomorphisms are there of Z into Z?
A subgroup H is conjugate to a subgroup K of a group G if there exists an inner automorphism ig of G such that ig[H] = K. Show that conjugacy is an equivalence relation on the collection of subgroups
How many homomorphisms are there of Z into Z2?
Characterize the normal subgroups of a group G in terms of the cells where they appear in the partition given by the conjugacy relation in the preceding exercise.
Give an example of a group G having no elements of finite order > 1 but having a factor group G/H, all of whose elements are of finite order.
Let Hand K be normal subgroups of a group G. Give an example showing that we may have H ≈ K while G/H is not isomorphic to G/K.
Let G be a group, and let g ∈ G. Let ∅g : G → G be defined by ∅g(x) = gxg-1 for x ∈ G. For which g ∈ G is ∅g a homomorphism?
Let H be a normal subgroup of a group G, and let m = (G : H). Show that am ∈ H for every a ∈ G.
Describe the center of every simple a. Abelian group b. Nonabelian group.
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 homomorphism is a map such that
Describe the commutator subgroup of every simple a. Abelian group b. Nonabelian group.
Show that an intersection of normal subgroups of a group G is again a normal subgroup of G.
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 ∅ : G → G' be a homomorphism of
Given any subset S of a group G, show that it makes sense to speak of the smallest normal subgroup that contains S.
Give a one-sentence synopsis of the proof of Theorem 15.9.Data from Theorem 15.9.A factor group of a cyclic group is cyclic. Proof Let G be cyclic with generator a, and let N be a normal
Mark each of the following true or false. ___ a. A,, is a normal subgroup of Sn .___ b. For any two groups G and G', there exists a homomorphism of G into G'. ___ c. Every homomorphism is a
Let G be a group. An element of G that can be expressed in the form aba-1b-1 for some a, b ∈ G is a commutator in G. The preceding exercise shows that there is a smallest normal subgroup C of a
Show that if a finite group G has exactly one subgroup H of a given order, then H is a normal subgroup of G.
Give an example of a nontrivial homomorphism ∅ for the given groups, if an example exists. If no such homomorphism exists, explain why that is so.∅ : Z12 → Z5
Give at most a two-sentence synopsis of the proof of Theorem 15.18.Data from Theorem 15.18.M is a maximal normal subgroup of G if and only if G/M is simple. Proof: Let M be a maximal normal
Show that if H and N are subgroups of a group G, and N is normal in G, then H ∩ N is normal in H. Show by an example that H ∩ N need not be normal in G.
Give an example of a nontrivial homomorphism ∅ for the given groups, if an example exists. If no such homomorphism exists, explain why that is so.∅ : Z12 → Z4
Let ∅ : G → G' be a group homomorphism, and let N be a normal subgroup of G. Show that ∅[NJ is normal subgroup of ∅[G].
Give an example of a nontrivial homomorphism ∅ for the given groups, if an example exists. If no such homomorphism exists, explain why that is so.∅ : Z2 x Z4 → Z2 x Z5
Let G be a group containing at least one subgroup of a fixed finite order s. Show that the intersection of all subgroups of G of order s is a normal subgroup of G.
Let ∅ : G → G' be a group homomorphism, and let N' be a normal subgroup of G'. Show that ∅-1 [N'] is a normal subgroup of G.
Give an example of a nontrivial homomorphism ∅ for the given groups, if an example exists. If no such homomorphism exists, explain why that is so.∅: Z3 → Z
a. Show that all automorphisms of a group G form a group under function composition. b. Show that the inner automorphisms of a group G form a normal subgroup of the group of all automorphisms of
Show that if G is nonabelian, then the factor group G/Z(G) is not cyclic.
Give an example of a nontrivial homomorphism ∅ for the given groups, if an example exists. If no such homomorphism exists, explain why that is so.∅ : Z3 → S3
Show that the set of all g ∈ G such that ig : G → G is the identity inner automorphism ie is a normal subgroup of a group G.
Using Exercise 37, show that a nonabelian group G of order pq where p and q are primes has a trivial center. Data from Exercise 37.Show that if G is nonabelian, then the factor group G/Z(G) is
Give an example of a nontrivial homomorphism ∅ for the given groups, if an example exists. If no such homomorphism exists, explain why that is so.∅ : Z → S3
Let G and G' be groups, and let H and H' be normal subgroups of G and G', respectively. Let ∅ be a homomorphism of G into G'. Show that ∅ induces a natural homomorphism ∅* : (G/ H) → (G' /
Prove that An is simple for n ≥ 5, following the steps given. a. Show An contains every 3-cycle if n ≥ 3.b. Show An is generated by the 3-cycles for n ≥ 3.c. Let r and s be fixed elements
Give an example of a nontrivial homomorphism ∅ for the given groups, if an example exists. If no such homomorphism exists, explain why that is so.∅ : Z x Z → 2Z
Use the properties det(AB) = det(A) • det(B) and det(In) = 1 for n x n matrices to show the following: a. Then x n matrices with determinant 1 form a normal subgroup of GL(n, R). b. Then
Let N be a normal subgroup of G and let H be any subgroup of G. Let HN = { hn | h∈ H, n ∈ N}. Show that H N is a subgroup of G, and is the smallest subgroup containing both N and H.
Give an example of a nontrivial homomorphism ∅ for the given groups, if an example exists. If no such homomorphism exists, explain why that is so.∅ : 2Z → Z x Z
Let G be a group, and let P(G) be the set of all subsets of G. For any A, B ∈ P(G), let us define the product subset AB = {ab | a ∈ A, b ∈ B}.a. Show that this multiplication of subsets is
With reference to the preceding exercise, let M also be a normal subgroup of G. Show that NM is again a normal subgroup of G.Data from Exercise 40 Let N be a normal subgroup of G and let H be
Give an example of a nontrivial homomorphism ∅ for the given groups, if an example exists. If no such homomorphism exists, explain why that is so.∅ : D4 → S3
Show that if H and K are normal subgroups of a group G such that H ∩ K = {e}, then hk = kh for all h ∈ H and k ∈ K.
Give an example of a nontrivial homomorphism ∅ for the given groups, if an example exists. If no such homomorphism exists, explain why that is so.∅ : S3 → S4
Give an example of a nontrivial homomorphism ∅ for the given groups, if an example exists. If no such homomorphism exists, explain why that is so.∅ : S4 → S3
Show that if G, G', and G" are groups and if ∅ : G → G' and γ : G' → G" are homomorphisms, then the composite map γ∅ : G → G" is a homomorphism.
Let ∅ : G → H be a group homomorphism. Show that ∅[G] is abelian if and only if for all x, y ∈ G, we have xyx-1 y-1 ∈ Ker(∅).
Let G be any group and let a be any element of G. Let ∅ : Z → G be defined by ∅(n) =an. Show that ∅ is a homomorphism. Describe the image and the possibilities for the kernel of ∅.
Let G be a group, Let h, k ∈ G and let ∅: Z x Z → G be defined by ∅(m, n) = hmkn. Give a necessary and sufficient condition, involving hand k, for ∅ to be a homomorphism. Prove your
Let ∅ G → G' be a homomorphism with kernel H and let a ∈ G. Prove the set equality {x ∈ G| ∅(x) = ∅(a)}= Ha.
Find a necessary and sufficient condition on G such that the map ∅ described in the preceding exercise is a homomorphism for all choices of h, k ∈ G.
Let G be a group, h an element of G, and n a positive integer. Let ∅ : Zn → G be defined by ∅(i) = hi for 0 ≤ i ≤ n. Give a necessary and sufficient condition (in terms of h and n) for ∅
Repeat Exercise 6 for the subgroup {Po, p2} of D4.Data from Exercise 6Find all left cosets of the subgroup {p0 , µ2} of the group D4 given by Table 8.12. Po PI P2 P3 12 8₁ Po 8₂ Po μ₁
Compute the indicated product of cycles that are permutations of { 1, 2, 3, 4, 5, 6, 7, 8}.(1, 3, 2, 7)(4, 8, 6)
Find all proper nontrivial subgroups of Z2 x Z2 •
For each of the four types of plane isometries ( other than the identity), give the possibilities for the order of an isometry of that type in the group of plane isometries.

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