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mathematics
a first course in abstract algebra

A First Course In Abstract Algebra 7th Edition John Fraleigh - Solutions

Find the number of irreducible quadratic polynomials in Zp[x], where p is a prime.
Give an example of a ring with unity 1 ≠ 0 that has a subring with nonzero unity 1' ≠ 1.
Mark each of the following true or false.___ a. Every field is also a ring. ___ b. Every ring has a multiplicative identity. ___ c. Every ring with unity has at least two units. ___ d. Every ring with unity has at most two units.___ e. It is possible for a subset of some field to be
Give a synopsis of the proof of Corollary 23.6. Data from 23.6 Corollary If G is a finite subgroup of the multiplicative group (F*, ·) of a field F, then G is cyclic. In particular, the multiplicative group of all nonzero elements of a finite field is cyclic. Proof: By Theorem 11.12
Show that for p a prime, the polynomial xP + a in Zp[x] is not irreducible for any a ∈ ZP.
If F is a field and a ≠ 0 is a zero of f (x) = a0 + a1x + • • • + anxn in F[x], show that 1/a is a zero of an +an-1x+··· +a0xn.
Show that the evaluation map ∅a of Example 18.10 satisfies the multiplicative requirement for a homomorphism.Data from in18.10 Example Let F be the ring of all functions mapping IR. into IR. defined in Example 18.4. For each a ∈ R, we have the evaluation homomorphism ∅a: F → R, where
Let f(x) ∈ F[x] where F is a field, and let α ∈ F. Show that the remainder r(x) when f(x) is divided by x - α, in accordance with the division algorithm, is f(α).
Let σm : Z → Zm be the natural homomorphism given by σm(a) = (the remainder of a when divided by m) for a ∈ Z. a. Show that σ̅m : Z[x] → Zm[x] given by σ̅m(a0 + a1x + · · · + anxn) = σm(a0) + σm(a1)x + · · · + σm(an)xn is a homomorphism of Z[x] onto Zm[x].b. Show that
Complete the argument outlined after Definitions 18.12 to show that isomorphism gives an equivalence relation on a collection of rings. Data from in Definition 18.12An isomorphism ∅ : R → R' from a ring R to a ring R' is a homomorphism that is one to one and onto R'. The rings R and R' are
Show that if U is the collection of all units in a ring (R, +,•)with unity, then (U, ·) is a group.
Show that a2 - b2 =(a+ b)(a - b) for all a and bin a ring R if and only if R is commutative.
Let (R, +) be an abelian group. Show that (R, +, ·) is a ring if we define ab = 0 for all a, b ∈ R.
Show that the rings 2Z and 3Z are not isomorphic. Show that the fields R and C are not isomorphic.
Let p be a prime. Show that in the ring Zp we have (a+ b)P = aP + bP for all a, b ∈ Zp.
Show that the multiplicative inverse of a unit a ring with unity is unique.
Show that the unity element in a subfield of a field must be the unity of the whole field, in contrast to Exercise 32 for rings. Data from Exercise 32Give an example of a ring with unity 1 ≠ 0 that has a subring with nonzero unity 1' ≠ 1.
Recall that for an m x n matrix A, the transpose AT of A is the matrix whose jth column is the jth row of A. Show that if A is an m x n matrix such that ATA is invertible, then the projection matrix P = A(ATA)-1AT is an idempotent in the ring of n x n matrices.
A ring R is a Boolean ring if a2 = a for all a ∈ R, so that every element is idempotent. Show that every Boolean ring is commutative.
Classify the given group according to the fundamental theorem of finitely generated abelian groups. (Z2 X Z4)/((0, 1))
Determine whether the given map ∅ is a homomorphism.Let ∅ : R→ R*, where R is additive and R* is multiplicative, be given by ∅(x) = 2x.
Repeat the preceding exercise, but find the right cosets this time. Are they the same as the left cosets?Data from Exercise 6 Find all left cosets of the subgroup {p0 , µ2} of the group D4 given by Table 8.12.Table 8.12 Po PI P2 P3 f 14₂ 8₁ 82 Po PI P₂ P3 μ| 12 8₁ Po Pi P2
Illustrate the seven different types of friezes when they are classified according to their symmetries. Imagine the figure shown to be continued infinitely to the right and left. The symmetry group of a frieze always contains translations. For each of these exercises answer these questions about
Illustrate the seven different types of friezes when they are classified according to their symmetries. Imagine the figure shown to be continued infinitely to the right and left. The symmetry group of a frieze always contains translations. For each of these exercises answer these questions about
Find the order of the given element of the direct product.(8, 10) in Z12 x Z18
Illustrate the seven different types of friezes when they are classified according to their symmetries. Imagine the figure shown to be continued infinitely to the right and left. The symmetry group of a frieze always contains translations. For each of these exercises answer these questions about
Illustrate the seven different types of friezes when they are classified according to their symmetries. Imagine the figure shown to be continued infinitely to the right and left. The symmetry group of a frieze always contains translations. For each of these exercises answer these questions about
Mark each of the following true or false. ___ a. Every permutation is a cycle. ___ b. Every cycle is a permutation. ___ c. The definition of even and odd permutations could have been given equally well before Theorem 9.15. ___ d. Every nontrivial subgroup H of S9 containing
Illustrate the seven different types of friezes when they are classified according to their symmetries. Imagine the figure shown to be continued infinitely to the right and left. The symmetry group of a frieze always contains translations. For each of these exercises answer these questions about
Illustrate the seven different types of friezes when they are classified according to their symmetries. Imagine the figure shown to be continued infinitely to the right and left. The symmetry group of a frieze always contains translations. For each of these exercises answer these questions about
Use Exercise 27 to determine the number of abelian groups (up to isomorphism) of order (10)5.Data from exercise 27Following the idea suggested in Exercise 26, let m and n be relatively prime positive integers. Show that if there are (up to isomorphism) r abelian groups of order m ands of order n,
Illustrate the seven different types of friezes when they are classified according to their symmetries. Imagine the figure shown to be continued infinitely to the right and left. The symmetry group of a frieze always contains translations. For each of these exercises answer these questions about
The part of the decomposition of G in Theorem 11.12 corresponding to the subgroups of prime-power order can also be written in the form Zm1 x Zm2 x · · · x Zmr where mi divides mi+1 for i = 1, 2, · · ·, r - 1. The numbers mi can be shown to be unique, and are the torsion coefficients of
A permutation matrix is one that can be obtained from an identity matrix by reordering its rows. If P is an n x n permutation matrix and A is any n x n matrix and C = PA, then C can be obtained from A by making precisely the same reordering of the rows of A as the reordering of the rows which
Deal with the concept of the torsion subgroup just defined.An abelian group is torsion free if e is the only element of finite order. Use Theorem 11.12 to show that every finitely generated abelian group is the internal direct product of its torsion subgroup and of a torsion-free subgroup. (Note
let X={l, 2, 3, 4, s1.s2,s3,s4,m1,m2,d1,d2,C,P1,P2,P3,P4} be the D4-set of Example 16.8 with action table in Table 16.10. Find the following, where G = D4.The fixed sets Xσ for each σ ∈ D4, that is, XPo, Xp1 , • • ·, Xδ2 Data from in Example 16.8Let G be the group D4 = {p0, p1, p2,
let X={l, 2, 3, 4, s1.s2,s3,s4,m1,m2,d1,d2,C,P1,P2,P3,P4} be the D4-set of Example 16.8 with action table in Table 16.10. Find the following, where G = D4. The isotropy subgroups Gx for each x ∈ X, that is, G1, G2, · · · , Gp3 , GP4Data from in Example 16.8Let G be the group D4 = {p0, p1, p2,
Let G, H, and K be finitely generated abelian groups. Show that if G x K is isomorphic to H x K, then G ≈ H.
Determine whether the given map ∅ is a homomorphism.Let ∅ : Z→ R under addition be given by ∅(n) = n.
In each of the following exercises use Corollary 17 .2 to work the problem, even though the answer might be obtained by more elementary methods. Find the number of orbits in {1, 2, 3, 4, 5, 6, 7, 8} under the cyclic subgroup ((1, 3, 5, 6)) of S8.Data from 17.2 Corollary If G is a finite
Find the order of the given factor group. Z6/(3)
In each of the following exercises use Corollary 17 .2 to work the problem, even though the answer might be obtained by more elementary methods. Find the number of orbits in {l, 2, 3, 4, 5, 6, 7, 8} under the subgroup of S8 generated by (1, 3) and (2, 4, 7).Data from 17.2 CorollaryIf G is a
Find the order of the given factor group.(Z4 x Z12) / ((2) x (2))
Classify the given group according to the fundamental theorem of finitely generated abelian groups.(Z2 X Z4)/((0, 2))
Determine whether the given map ∅ is a homomorphism.Let ∅ : R* → R* under multiplication be given by ∅(x) = |x |.
In each of the following exercises use Corollary 17 .2 to work the problem, even though the answer might be obtained by more elementary methods.Find the number of distinguishable tetrahedral dice that can be made using one, two, three, and four dots on the faces of a regular tetrahedron, rather
Find the order of the given factor group.(Z4 X Z2) / ((2, 1))
In each of the following exercises use Corollary 17 .2 to work the problem, even though the answer might be obtained by more elementary methods. Wooden cubes of the same size are to be painted a different color on each face to make children's blocks. How many distinguishable blocks can be made
let X={l, 2, 3, 4, s1.s2,s3,s4,m1,m2,d1,d2,C,P1,P2,P3,P4} be the D4-set of Example 16.8 with action table in Table 16.10. Find the following, where G = D4.The orbits in X under D4Data from in Example 16.8Let G be the group D4 = {p0, p1, p2, p3, μ1, μ2, δ1, δ2 } of symmetries of the square,
Classify the given group according to the fundamental theorem of finitely generated abelian groups.(Z2 X Z4)/((1, 2))
Classify the given group according to the fundamental theorem of finitely generated abelian groups.(Z4 x Z8)/((1, 2))
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 acts faithfully on X if and only if gx = x implies that g = e.
Find the order of the given factor group.(Z3 x Z5)/({0} x Z5)
Determine whether the given map ∅ is a homomorphism.Let ∅ : Z6 → Z2 be given by ∅(x) = the remainder of x when divided by 2, as in the division algorithm.
Find the order of the given factor group.(Z2 x Z4) / ((l, 1))
Determine whether the given map ∅ is a homomorphism.Let ∅ : Z9 → Z2 be given by ∅(x) = the remainder of x when divided by 2, as in the division algorithm.
Classify the given group according to the fundamental theorem of finitely generated abelian groups.(Z4 x Z4 x Z8)/((1, 2, 4))
Answer Exercise 4 if colors may be repeated on different faces at will. Data from exercise 4Wooden cubes of the same size are to be painted a different color on each face to make children's blocks. How many distinguishable blocks can be made if eight colors of paint are available?
Find the order of the given factor group.(Z12 x Z18) / ((4, 3))
Let X be a G-set and let S ⊆ X. If Gs ⊆ S for all s ∈ S, then S is a sub-G-set. Characterize a sub-G-set of a G-set X in terms of orbits in X and G.
Each of the eight comers of a cube is to be tipped with one of four colors, each of which may be used on from one to all eight comers. Find the number of distinguishable markings possible.
Classify the given group according to the fundamental theorem of finitely generated abelian groups.(Z x Z)/((0, 1))
Determine whether the given map ∅ is a homomorphism.Let ∅ G → G1 x G2 x • • • x Gi x · · · x Gr. be given by ∅i(gi) = (e 1 • e2 .... , gi, ... , er.), where gi ∈ Gi and ej is the identity element of Gj. This is an injection map. Compare with Example 13.8. Data from Example
Find the order of the given factor group.(Z2 x S3) / ((l, P1))
Characterize a transitive G-set in terms of its orbits.
Classify the given group according to the fundamental theorem of finitely generated abelian groups.(Z x Z)/((1, 2))
Determine whether the given map ∅ is a homomorphism.Let G be any group and let ∅ : G → G be given by ∅(g) = g-1 for g ∈ G.
Find the order of the given factor group.(Z11 x Z15) / ((1, 1))
Classify the given group according to the fundamental theorem of finitely generated abelian groups.(Z x Z x Z)/((1, 1, 1))
Mark each of the following true or false. ___ a. Every G-set is also a group. ___ b. Each element of a G-set is left fixed by the identity of G. ___ c. If every element of a G-set is left fixed by the same element g of G, then g must be the identity e.___ d. Let X be a G-set with x1,
Consider six straight wires of equal lengths with ends soldered together to form edges of a regular tetrahedron. Either a 50-ohm or 100-ohm resistor is to be inserted in the middle of each wire. Assume there are at least six of each type of resistor available. How many essentially different wirings
Determine whether the given map ∅ is a homomorphism.Let F be the additive group of functions mapping R into R having derivatives of all orders. Let ∅ :F → F be given by ∅(f) = f", the second derivative of f.
Give the order of the element in the factor group.5 + (4) in Z12/ (4)
Classify the given group according to the fundamental theorem of finitely generated abelian groups.(Z x Z X Z4)/((3, 0, 0))
A rectangular prism 2ft long with 1-ft square ends is to have each of its six faces painted with one of six possible colors. How many distinguishable painted prisms are possible if a. No color is to be repeated on different faces, b. Each color may be used on any number of faces?
Let X be the D4-set in Example 16.8. a. Does D4 act faithfully on X? b. Find all orbits in X on which D4 acts faithfully as a sub-D4-set.Data from in Example 16.8Let G be the group D4 = {p0, p1, p2, p3, μ1, μ2, δ1, δ2 } of symmetries of the square, described. In Fig. 16.9 we show the
Determine whether the given map ∅ is a homomorphism.Let F be the additive group of all continuous functions mapping R into R. Let R be the additive group of real numbers, and let ∅ : F → R be given by $(f) = S f(x)dx.
Let X and Y be G-sets with the same group G. An isomorphism between G-sets X and Y is a map ∅ : X → Y that is one to one, onto Y, and satisfies g∅(x) = ∅(gx) for all x ∈ X and g ∈ G. Two G-sets are isomorphic if such an isomorphism between them exists. Let X be the D4-set of Example
Give the order of the element in the factor group.26+(12)inZ60/ (12)
Classify the given group according to the fundamental theorem of finitely generated abelian groups.(Z x Z x Z8)/ ((0. 4, 0))
Let X be a G-set. Show that G acts faithfully on X if and only if no two distinct elements of G have the same action on each element of X.
Determine whether the given map ∅ is a homomorphism.Let F be the additive group of all functions mapping R into R, and let ∅ : F → F be given by ∅(f) = 3f.
Classify the given group according to the fundamental theorem of finitely generated abelian groups.(Z x Z)/((2, 2))
Give the order of the element in the factor group.(2, 1) + ((1, 1)) in (Z3 X Z6)/((1, 1))
Let X be a G-set and let Y ⊆ X. Let Gy = {g ∈ G| gy = y for all y ∈ Y}. Show Gy is a subgroup of G, generalizing Theorem 16.12.Data from16.12 TheoremLet X be a G-set. Then Gx is a subgroup of G for each x ∈ X,Proof Let x ∈ X and let g1, g2 ∈ Gx. Then g1x = x and g2x = x. Consequently,
Determine whether the given map ∅ is a homomorphism.Let Mn be the additive group of all n x n matrices with real entries, and let R be the additive group of real numbers. Let ∅(A) = det(A), the determinant of A, for A ∈ Mn.
Give the order of the element in the factor group.3, 1) + ((1, 1)) in (Z4 X Z4)/ ((1, 1))
Classify the given group according to the fundamental theorem of finitely generated abelian groups.(Z x Z x Z)/((3, 3, 3))
Let G be the additive group of real numbers. Let the action of θ ∈ G on the real plane R2 be given by rotating the plane counterclockwise about the origin through θ radians. Let P be a point other than the origin in the plane. a. Show R2 is a G-set. b. Describe geometrically the orbit
Determine whether the given map ∅ is a homomorphism.Let Mn and R be as in Exercise 12. Let ∅(A)= tr(A) for A ∈ Mn, where the trace tr(A) is the sum of the elements on the main diagonal of A, from the upper-left to the lower-right corner.Data from Exercise 12Let Mn be the additive group of all
Give the order of the element in the factor group.(3, 1) + ((0, 2)) in (Z4 x Z8) / ((0, 2))
Show how all possible G-sets, up to isomorphism (see Exercise 9), can be formed from the group G.Let {Xi |i ∈ I} be a disjoint collection of sets, so Xi ∩ Xj = ∅ for i ≠ j. Let each Xi be a G-set for the same group G. a. Show that UiEIXi can be viewed in a natural way as a G-set, the
Determine whether the given map ∅ is a homomorphism.Let GL(n, R) be the multiplicative group of invertible n x n matrices, and let R be the additive group of real numbers. Let ∅: G L(n, R) → R be given by ∅(A)= tr(A), where tr(A) is defined in Exercise 13.Data from Exercise 13Let Mn and R
Give the order of the element in the factor group.(3, 3) + ((1, 2)) in (Z4 x Z8) / ((1, 2))
Find both the center and the commutator subgroup of Z3 x S3.
Show how all possible G-sets, up to isomorphism (see Exercise 9), can be formed from the group G.Let X be a transitive G-set, and let x0 ∈ X. Show that X is isomorphic (see Exercise 9) to the G-set L of all left cosets of Gx0 , described in Example 16.7.Data from in Example 16.7Let H be a
Determine whether the given map ∅ is a homomorphism.Let F be the multiplicative group of all continuous functions mapping R into R that are nonzero at every x ∈ R Let R* be the multiplicative group of nonzero real numbers. Let ∅ : F → R* be given by ∅(f)No, it is not a homomorphism. Let
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¢.Ker (∅) for ∅ : S3 →Z2 in Example 13.3Data from Example 13.3Let Sn be the symmetric group on n letters, and let ∅: Sn → Z2 be defined byShow that ∅ is a homomorphism.Solution: We must show that ∅(σµ) = ∅(σ) +
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, 2, 3}. We list the permutations of A and assign to each a subscripted Greek letter for a name.The

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