Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Chapter 6, Problem 45E
To determine
To show: The inner automorphism induced by
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Q.1) Classify the following statements as a true or false statements:
a. If M is a module, then every proper submodule of M is contained in a maximal
submodule of M.
b. The sum of a finite family of small submodules of a module M is small in M.
c. Zz is directly indecomposable.
d. An epimorphism a: M→ N is called solit iff Ker(a) is a direct summand in M.
e. The Z-module has two composition series.
Z
6Z
f. Zz does not have a composition series.
g. Any finitely generated module is a free module.
h. If O→A MW→ 0 is short exact sequence then f is epimorphism.
i. If f is a homomorphism then f-1 is also a homomorphism.
Maximal C≤A if and only if is simple.
Sup
Q.4) Give an example and explain your claim in each case:
Monomorphism not split.
b) A finite free module.
c) Semisimple module.
d) A small submodule A of a module N and a homomorphism op: MN, but
(A) is not small in M.
I need diagram with solutions
Chapter 6 Solutions
Contemporary Abstract Algebra
Ch. 6 - Prob. 1ECh. 6 - Find Aut(Z).Ch. 6 - Let R+ be the group of positive real numbers under...Ch. 6 - Show that U(8) is not isomorphic to U(10).Ch. 6 - Show that U(8) is isomorphic to U(12).Ch. 6 - Prove that isomorphism is an equivalence relation....Ch. 6 - Prove that S4 is not isomorphic to D12 .Ch. 6 - Show that the mapping alog10a is an isomorphism...Ch. 6 - In the notation of Theorem 6.1, prove that Te is...Ch. 6 - Given that is a isomorphism from a group G under...
Ch. 6 - Let G be a group under multiplication, G be a...Ch. 6 - Let G be a group. Prove that the mapping (g)=g1...Ch. 6 - Prob. 13ECh. 6 - Find two groups G and H such that GH , but...Ch. 6 - Prob. 15ECh. 6 - Find Aut(Z6) .Ch. 6 - If G is a group, prove that Aut(G) and Inn(G) are...Ch. 6 - If a group G is isomorphic to H, prove that Aut(G)...Ch. 6 - Suppose belongs to Aut(Zn) and a is relatively...Ch. 6 - Let H be the subgroup of all rotations in Dn and...Ch. 6 - Let H=S5(1)=1andK=S5(2)=2 . Provethat H is...Ch. 6 - Show that Z has infinitely many subgroups...Ch. 6 - Prob. 23ECh. 6 - Let be an automorphism of a group G. Prove that...Ch. 6 - Prob. 25ECh. 6 - Suppose that :Z20Z20 is an automorphismand (5)=5 ....Ch. 6 - Identify a group G that has subgroups isomorphic...Ch. 6 - Prove that the mapping from U(16) to itself given...Ch. 6 - Let rU(n) . Prove that the mapping a: ZnZn defined...Ch. 6 - The group {[1a01]|aZ} is isomorphic to what...Ch. 6 - If and are isomorphisms from the cyclic group a...Ch. 6 - Prob. 32ECh. 6 - Prove property 1 of Theorem 6.3. Theorem 6.3...Ch. 6 - Prove property 4 of Theorem 6.3. Theorem 6.3...Ch. 6 - Referring to Theorem 6.1, prove that Tg is indeed...Ch. 6 - Prove or disprove that U(20) and U(24) are...Ch. 6 - Show that the mapping (a+bi)=a=bi is an...Ch. 6 - Let G={a+b2a,barerational} and...Ch. 6 - Prob. 39ECh. 6 - Explain why S8 contains subgroups isomorphic to...Ch. 6 - Let C be the complex numbers and M={[abba]|a,bR} ....Ch. 6 - Prob. 42ECh. 6 - Prob. 43ECh. 6 - Suppose that G is a finite Abelian group and G has...Ch. 6 - Prob. 45ECh. 6 - Prob. 46ECh. 6 - Suppose that g and h induce the same inner...Ch. 6 - Prob. 48ECh. 6 - Prob. 49ECh. 6 - Prob. 50ECh. 6 - Prob. 51ECh. 6 - Let G be a group. Complete the following...Ch. 6 - Suppose that G is an Abelian group and is an...Ch. 6 - Let be an automorphismof D8 . What are the...Ch. 6 - Let be an automorphism of C*, the group of...Ch. 6 - Let G=0,2,4,6,...andH=0,3,6,9,... .Prove that G...Ch. 6 - Give three examples of groups of order 120, no two...Ch. 6 - Let be an automorphism of D4 such that (H)=D ....Ch. 6 - Prob. 59ECh. 6 - Prob. 60ECh. 6 - Write the permutation corresponding to R90 in the...Ch. 6 - Show that every automorphism of the rational...Ch. 6 - Prove that Q+ , the group of positive rational...Ch. 6 - Prob. 64ECh. 6 - Prob. 65ECh. 6 - Prove that Q*, the group of nonzero rational...Ch. 6 - Give a group theoretic proof that Q under addition...
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- T. Determine the least common denominator and the domain for the 2x-3 10 problem: + x²+6x+8 x²+x-12 3 2x 2. Add: + Simplify and 5x+10 x²-2x-8 state the domain. 7 3. Add/Subtract: x+2 1 + x+6 2x+2 4 Simplify and state the domain. x+1 4 4. Subtract: - Simplify 3x-3 x²-3x+2 and state the domain. 1 15 3x-5 5. Add/Subtract: + 2 2x-14 x²-7x Simplify and state the domain.arrow_forwardQ.1) Classify the following statements as a true or false statements: Q a. A simple ring R is simple as a right R-module. b. Every ideal of ZZ is small ideal. very den to is lovaginz c. A nontrivial direct summand of a module cannot be large or small submodule. d. The sum of a finite family of small submodules of a module M is small in M. e. The direct product of a finite family of projective modules is projective f. The sum of a finite family of large submodules of a module M is large in M. g. Zz contains no minimal submodules. h. Qz has no minimal and no maximal submodules. i. Every divisible Z-module is injective. j. Every projective module is a free module. a homomorp cements Q.4) Give an example and explain your claim in each case: a) A module M which has a largest proper submodule, is directly indecomposable. b) A free subset of a module. c) A finite free module. d) A module contains no a direct summand. e) A short split exact sequence of modules.arrow_forwardListen ANALYZING RELATIONSHIPS Describe the x-values for which (a) f is increasing or decreasing, (b) f(x) > 0 and (c) f(x) <0. y Af -2 1 2 4x a. The function is increasing when and decreasing whenarrow_forwardBy forming the augmented matrix corresponding to this system of equations and usingGaussian elimination, find the values of t and u that imply the system:(i) is inconsistent.(ii) has infinitely many solutions.(iii) has a unique solutiona=2 b=1arrow_forwardif a=2 and b=1 1) Calculate 49(B-1)2+7B−1AT+7ATB−1+(AT)2 2)Find a matrix C such that (B − 2C)-1=A 3) Find a non-diagonal matrix E ̸= B such that det(AB) = det(AE)arrow_forwardWrite the equation line shown on the graph in slope, intercept form.arrow_forward1.2.15. (!) Let W be a closed walk of length at least 1 that does not contain a cycle. Prove that some edge of W repeats immediately (once in each direction).arrow_forward1.2.18. (!) Let G be the graph whose vertex set is the set of k-tuples with elements in (0, 1), with x adjacent to y if x and y differ in exactly two positions. Determine the number of components of G.arrow_forward1.2.17. (!) Let G,, be the graph whose vertices are the permutations of (1,..., n}, with two permutations a₁, ..., a,, and b₁, ..., b, adjacent if they differ by interchanging a pair of adjacent entries (G3 shown below). Prove that G,, is connected. 132 123 213 312 321 231arrow_forward1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k components, where k is the greatest common divisor of {n, r,s}.arrow_forward1.2.20. (!) Let u be a cut-vertex of a simple graph G. Prove that G - v is connected. עarrow_forward1.2.12. (-) Convert the proof at 1.2.32 to an procedure for finding an Eulerian circuit in a connected even graph.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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