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mathematics
a first course in abstract algebra

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

Show that the number of conjugate classes in Sn is also the number of different abelian groups (up to isomorphism) of order pn, where p is a prime number.
Let G be a finitely generated abelian group with identity 0. A finite set {b1, · · · , bn}, where bi ∈ G, is a basis for G if {b1, · · · , bn} generates G and ∑ni=1 =1 mibi = 0 if and only if each mibi = 0, where mi ∈ Z. a. Show that {2, 3} is not a basis for Z4 . Find a basis
Let H be a subgroup of a group G. Show that GH = {g ∈ G| gHg-1 = H} is a subgroup of G.
Show that free abelian groups of finite rank are precisely the finitely generated abelian groups containing no nonzero elements of finite order.
We showed in Example 15.6 that A4 has no subgroup of order 6. The preceding exercise shows that such a subgroup of A4 would have to be isomorphic to either Z6 or S3. Show again that this is impossible by considering orders of elements.Data from in Example 15.6The theorem of Lagrange states if H is
Let G be any group. An abelian group G* is a blip group of G if there exists a fixed homomorphism ∅ of G onto G* such that each homomorphism ψ of G into an abelian group G' can be factored as ψ = θ∅, where θ is a homomorphism of G* into G' (see Fig. 39.14). a. Show that any two blip
Find the center of S3 x D4.
Show that if n > 2, the center of Sn is the subgroup consisting of the identity permutation only.
Let S be any set. A group G together with a fixed function g : S → G constitutes a blop group on S if for each group G' and map f : S → G' there exists a unique homomorphism ∅f of G into G' such that f = ∅f g (see Fig. 39.15). a. Let S be a fixed set. Show that if both G1,
Let G be a finite group and let primes p and q ≠ p divide |G|. Prove that if G has precisely one proper Sylow p-subgroup, it is a normal subgroup, so G is not simple.
Show that Q under addition is not a free abelian group.
Let S = {aibj|0 ≤ i < m, 0 ≤ j < n}, that is, S consists of all formal products aibj starting with a0b0 and ending with am-1bn-1. Let r be a positive integer, and define multiplication on S by (asbt)(aubv) = axby, where x is the remainder of s + u(rt) when divided by m, and y is the
Find the ascending central series of S3 x Z4.
Characterize a free abelian group by properties in a fashion similar to that used in Exercise 13. Data from Exercise 13Let S be any set. A group G together with a fixed function g : S → G constitutes a blop group on S if for each group G' and map f : S → G' there exists a unique
Show that every group of order 45 has a normal subgroup of order 9.
Deal with showing the uniqueness of the prime powers appearing in the prime-power decomposition of the torsion subgroup T of a finitely generated abelian group. Let p be a fixed prime. Show that the elements of T having as order some power of p, together with zero, form a subgroup Tp of T.
Find the ascending central series of S3 x D4.
Show that if n = pq, with p and q primes and q > p and q ≡ 1 (mod p ), then there is exactly one nonabelian group (up to isomorphism) of order n. Recall that the q - 1 nonzero elements of Zq form a cyclic group Zq* under multiplication modulo q.
Prove Corollary 36.4. Data from Corollary 36.4Let G be a finite group. Then G is a p-group if and only if |G| is a power of p. Proof We leave the proof of this corollary to Exercise 14.
Deal with showing the uniqueness of the prime powers appearing in the prime-power decomposition of the torsion subgroup T of a finitely generated abelian group.Show that in any prime-power decomposition of T, the subgroup TP in the preceding exercise is isomorphic to the direct product of those
Deal with showing the uniqueness of the prime powers appearing in the prime-power decomposition of the torsion subgroup T of a finitely generated abelian group.Using Exercise 17, show that provided each ri ≥ 1.Data from Exercise 17Referring to Exercise 16, show that Zpr [p] ≈ ZP for any r
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 composition series of a group G is a finite sequence {e} = H0 < H1 < H2 < · · · < Hn-1 < Hn = G of subgroups of G such that Hi
Let G be a finite group and let p be a prime dividing |G|. Let P be a Sylow p-subgroup of G. Show that N[N[P]] = N[P].
Deal with showing the uniqueness of the prime powers appearing in the prime-power decomposition of the torsion subgroup T of a finitely generated abelian group.Let G be any abelian group and let n be any positive integer. Show that G[n] = {x ∈ G | nx = 0} is a subgroup of G. (In multiplicative
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 solvable group is one that has a composition series of abelian groups.
Is the group D4 of symmetries of the square in Example 8.10 solvable?Data from in Example 8.10Let us form the dihedral group D4 of permutations corresponding to the ways that two copies of a square with vertices 1, 2, 3, and 4 can be placed, one covering the other with vertices on top of vertices
Let G be a finite group and let a prime p divide |G|. Let P be a Sylow p-subgroup of G and let H be any p-subgroup of G. Show there exists g ∈ G such that gHg-1 ≤ P.
Deal with showing the uniqueness of the prime powers appearing in the prime-power decomposition of the torsion subgroup T of a finitely generated abelian group.Referring to Exercise 16, show that Zpr [p] ≈ ZP for any r ≥ 1 and prime p.Data from Exercise 16Let G be any abelian group and let n be
Mark each of the following true or false. ___ a. Every normal series is also subnormal. ___ b. Every subnormal series is also normal. ___ c. Every principal series is a composition series. ___ d. Every composition series is a principal series. ___ e. Every abelian group has
Show that every group of order (35)3 has a normal subgroup of order 125.
Find a composition series of S3 x S3. Is S3 x S3 solvable?
Let G be Z36 . Refer to the proof of Theorem 35.11. Let the subnormal series (1) be {0} < (12) < (3) < Z36 and let the subnormal series (2) be {0} < (18) < Z36. Find chains (3) and (4) and exhibit the isomorphic factor groups as described in the proof. Write chains (3) and (4) in the
Show that there are no simple groups of order 255 = (3)(5)(17).
Deal with showing the uniqueness of the prime powers appearing in the prime-power decomposition of the torsion subgroup T of a finitely generated abelian group. Let G be a finitely generated abelian group and Tp the subgroup defined in Exercise 14. Suppose Tp ≈ Zpr1 x Zpr2 X ... X Zprm ≈
Show that there are no simple groups of order prm, where pi s a prime, r is a positive integer, and m < p.
Let T be the torsion subgroup of a finitely generated abelian group. Suppose T ≈ Zm1 x Zm2 x · · · x Zmr ≈ Zn1, x Zn2 x · · · x Zns, where mi divides mi+1 for i = 1, · · ·, r - 1, and nj divides nj+1 for n = 1, · · ·, s - 1, and m1 > 1 and n1 > 1. We wish to show that r = s
Let G be a finite group. Regard G as a G-set where G acts on itself by conjugation. a. Show that GG is the center Z(G) of G. b. Use Theorem 36.1 to show that the center of a finite nontrivial p-group is nontrivial.Data from Theorem 36.1Let G be a group of order pn and let X be a finite
Let T be the torsion subgroup of a finitely generated abelian group. Suppose T ≈ Zm1 x Zm2 x · · · x Zmr ≈ Zn1, x Zn2 x · · · x Zns, where mi divides mi+1 for i = 1, · · ·, r - 1, and nj divides nj+1 for n = 1, · · ·, s - 1, and m1 > 1 and n1 > 1. We wish to show that r = s
Repeat Exercise 20 for the group Z24 with the subnormal series (1) {0} < (12) < (4) < Z24 and (2) {0} < (6) < (3) < Z24.Data from Exercise 20Let G be Z36 . Refer to the proof of Theorem 35.11. Let the subnormal series (1) be {0} < (12) < (3) < Z36 and let the subnormal
Let p be a prime. Show that a finite group of order pn contains normal subgroups Hi for 0 ≤ i ≤ n such that |Hi| = pi and Hi < Hi+1 for 0 ≤ i < n.
Let T be the torsion subgroup of a finitely generated abelian group. Suppose T ≈ Zm1 x Zm2 x · · · x Zmr ≈ Zn1, x Zn2 x · · · x Zns, where mi divides mi+1 for i = 1, · · ·, r - 1, and nj divides nj+1 for n = 1, · · ·, s - 1, and m1 > 1 and n1 > 1. We wish to show that r = s
Let H*, H, and K be subgroups of G with H* normal in H. Show that H* ∩ K is normal in H ∩ K.
Let G be a finite group and let P be a normal p-subgroup of G. Show that P is contained in every Sylow p-subgroup of G.
Show that if H0= {e} < H1 < H2 < · · · < Hn = G is a subnormal (normal) series for a group G, and if Hi+1/Hi is of finite order Si+1, then G is of finite order S1S2 ···Sn.
Show that an infinite abelian group can have no composition series.
Show that a finite direct product of solvable groups is solvable.
Show that a subgroup K of a solvable group G is solvable.
Let H0 = {e} < H1 < ··· < Hn = G be a composition series for a group G. Let N be a normal subgroup of G, and suppose that N is a simple group. Show that the distinct groups among H0, HiN for i = 0,··· , n also form a composition series for G.
Let G be a group, and let H0 = {e} < H1 < ··· < Hn = G be a composition series for G. Let N be a normal subgroup of G, and let γ : G → G/N be the canonical map. Show that the distinct groups among γ[Hi] for i = 0, ··· , n, form a composition series for G/N.
Prove that a homomorphic image of a solvable group is solvable.
Show that the given number α ∈ C is algebraic over Q by finding f(x) ∈ Q[x] such that f(α) = 0.1 + √2
Determine whether there exists a finite field having the given number of elements.4096
Prove the trigonometric identity cos3θ = 4 cos3θ - 3 cosθ from the Euler formula, eiθ = cosθ + i sin θ.
Find the degree and a basis for the given field extension. Be prepared to justify your answers.Q(√2) over Q
Find three bases for R2 over R, no two of which have a vector in common.
Using the proof of Theorem 32.11, show that the regular 9-gon is not constructible. Data from Theorem 32.11Trisecting the angle is impossible; that is, there exists an angle that cannot be trisected with a straightedge and a compass. Proof Figure 32.12 indicates that the angle θ can be
Determine whether there exists a finite field having the given number of elements.3127
Find the degree and a basis for the given field extension. Be prepared to justify your answers.Q(√2,√3) over Q
Determine whether the given set of vectors is a basis for R3 over R.{(1, 1, 0), (1, 0, 1), (0, 1, 1)}
Determine whether there exists a finite field having the given number of elements.68,921
Find the degree and a basis for the given field extension. Be prepared to justify your answers.Q(√2, √3,√18) over Q
Show that the given number α ∈ C is algebraic over Q by finding f(x) ∈ Q[x] such that f(α) = 0. +泛
Show that the given number α ∈ C is algebraic over Q by finding f(x) ∈ Q[x] such that f(α) = 0. √√√√2-i
Determine whether the given set of vectors is a basis for R3 over R.{(-1, 1, 2), (2, -3, 1), (10, -14, 0)}
Find the number of primitive 8th roots of unity in GF(9).
Referring to Fig. 32.13, where AQ̅ bisects angle OAP, show that the regular 10-gon is constructible (and therefore that the regular pentagon is also). 36° 1 1 Q 32.13 Figure A P
Find the degree and a basis for the given field extension. Be prepared to justify your answers.Q(3√2, √3) over Q
Show algebraically that it is possible to construct an angle of 30°.
Give a basis for the indicated vector space over the field.Q(√2) over Q
Find irr(α, Q) and deg(α, Q) for the given algebraic number a ∈ C Be prepared to prove that your polynomials are irreducible over Q if challenged to do so. 3-√√6
Find the number of primitive 18th roots of unity in GF(19).
Find the degree and a basis for the given field extension. Be prepared to justify your answers.Q(√2, 3√2) over Q
Give a basis for the indicated vector space over the field.R(√2) over R
Use the results of Exercise 5 where needed to show that the statement is true.The regular 20-gon is constructible. Data from Exercise 5Referring to Fig. 32.13, where AQ̅ bisects angle OAP, show that the regular 10-gon is constructible (and therefore that the regular pentagon is also).
Find the number of primitive 15th roots of unity in GF(31).
Find irr(α, Q) and deg(α, Q) for the given algebraic number a ∈ C Be prepared to prove that your polynomials are irreducible over Q if challenged to do so. √ ( 3 ) +√7
Find the degree and a basis for the given field extension. Be prepared to justify your answers.Q(√2 + √3) over Q
Use the results of Exercise 5 where needed to show that the statement is true.The regular 30-gon is constructible. Data from Exercise 5Referring to Fig. 32.13, where AQ̅ bisects angle OAP, show that the regular 10-gon is constructible (and therefore that the regular pentagon is also).
Give a basis for the indicated vector space over the field.Q(3√2) over Q
Find the number of primitive 10th roots of unity in GF(23).
Find the degree and a basis for the given field extension. Be prepared to justify your answers.Q(√2√3) over Q
Use the results of Exercise 5 where needed to show that the statement is true.The angle 72° can be trisected.Data from Exercise 5Referring to Fig. 32.13, where AQ̅ bisects angle OAP, show that the regular 10-gon is constructible (and therefore that the regular pentagon is also). 36° 1 1 Q 32.13
Give a basis for the indicated vector space over the field.C over R
Find irr(α, Q) and deg(α, Q) for the given algebraic number a ∈ C Be prepared to prove that your polynomials are irreducible over Q if challenged to do so.√2 + i
Mark each of the following true or false. ___ a. The nonzero elements of every finite field form a cyclic group under multiplication. ___ b. The elements of every finite field form a cyclic group under addition. ___ c. The zeros in C of (x28 - 1) ∈ Q[x] form a cyclic group under
Find the degree and a basis for the given field extension. Be prepared to justify your answers.Q(√2,3√5) over Q
Use the results of Exercise 5 where needed to show that the statement is true.The regular 15-gon can be constructed.Data from Exercise 5Referring to Fig. 32.13, where AQ̅ bisects angle OAP, show that the regular 10-gon is constructible (and therefore that the regular pentagon is also).
Classify the given a ∈ C as algebraic or transcendental over the given field F. If α is algebraic over F, find deg(α, F).α = i, F = Q
Let Z̅2 be an algebraic closure of Z2, and let α, β ∈ Z̅2 be zeros of x3 + x2 + 1 and of x3 + x2 + 1, respectively. Using the results of this section, show that Z2(α) = Z2(β).
Find the degree and a basis for the given field extension. Be prepared to justify your answers.Q(3√2, 3√6,3√24) over Q
Give a basis for the indicated vector space over the field.Q(4√2) over Q
Classify the given a ∈ C as algebraic or transcendental over the given field F. If α is algebraic over F, find deg(α, F).
Show that every irreducible polynomial in Zp[x] is a divisor of xPn - x for some n.
Find the degree and a basis for the given field extension. Be prepared to justify your answers.Q(√2,√6) over Q(√3)
Suppose you wanted to explain roughly in just three or four sentences, for a high school plane geometry teacher who never had a course in abstract algebra, how it can be shown that it is impossible to trisect an angle of 60°. Write down what you would say.
According to Theorem 30.23, the element 1 + α of Z2(a) of Example 29.19 is algebraic over Z2. Find the irreducible polynomial for 1 + α in Z2[x].Data from in Example 29.19The polynomial p(x) = x2 + x + 1 in Z2[x] is irreducible over Z2 by Theorem 23.10, since neither element 0 nor element 1 of Z2
Classify the given a ∈ C as algebraic or transcendental over the given field F. If α is algebraic over F, find deg(α, F).
Let F be a finite field of pn elements containing the prime subfield Zp. Show that if α ∈ F is a generator of the cyclic group (F*, ·) of nonzero elements of F, then deg(α, Zp) = n.
Find the degree and a basis for the given field extension. Be prepared to justify your answers.Q(√2 + √3) over Q(√3)
Classify the given a ∈ C as algebraic or transcendental over the given field F. If α is algebraic over F, find deg(α, F).α = √π, F = R

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