1. The formula for a lincar transformation T is given below. Determine if the transfor-mation is one-to-one, onto, both, or neither.(a) " (E) -A(E)-)(E)-E%3D[1 + r2(b) Т%3D(c) T%3D

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Answered: 1. The formula for a linear…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.2: Cayley’s Theorem

Problem 12E: Find the right regular representation of G as defined Exercise 11 for each of the following groups....

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In Example 3, the group S(A) is nonabelian where A={ 1,2,3 }. Exhibit a set A such that S(A) is abelian. Example 3. We shall take A={ 1,2,3 } and obtain an explicit example of S(A). In order to define an element f of S(A), we need to specify f(1), f(2), and f(3). There are three possible choices for f(1). Since f is to be bijective, there are two choices for f(2) after f(1) has been designated, and then only once choice for f(3). Hence there are 3!=321 different mappings f in S(A).
11. Show that is a generating set for the additive abelian group if and only if
11. Assume that are subgroups of the abelian group such that the sum is direct. If is a subgroup of for prove that is a direct sum.
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Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)
Find the right regular representation of G as defined Exercise 11 for each of the following groups. a. G={ 1,i,1,i } from Example 1. b. The octic group D4={ e,,2,3,,,, }.
31. (See Exercise 30.) Prove that if and are primes and is a nonabelian group of order , then the center of is the trivial subgroup . Exercise 30: 30. Let be a group with center . Prove that if is cyclic, then is abelian.
1. Consider , the groups of units in under multiplication. For each of the following subgroups in , partition into left cosets of , and state the index of in a. b.
13. Assume that are subgroups of the abelian group . Prove that if and only if is generated by
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5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:

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1. The formula for a linear transformation T is given below. Determine if the transfor- mation is one-to-one, onto, both, or neither. (a) 7 (E) -A E)- (0) 7 (E) -A T. %3D (b) Т %3D x2 T