33. If a, bɛR with a # 0, let T:R→ R be the function given by T(x) = ax + b. Prove that the set G = {Tab|4, bER with a # 0} forms a nonabelian group under composition of functions. 34. Let H = {TlbeR} (notation as in Exercise 33). Prove that H is an abelian

Advanced Engineering Mathematics
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Thomas W. Hungerford - Abstrac x
b My Questions | bartleby
O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(201..
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of 621
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31. Let T be a set with at least three elements. Show that the permutation group
A(T) (Exercise 12) is nonabelian.
32. Let T be an infinite set and let A(T) be the group of permutations of T
(Exercise 12). Let M = {f€ A(T)| f(t) # t for only a finite number of teT}.
Prove that M is a group.
33. If a, bER with a # 0, let T:R→R be the function given by Tab(x) = ax + b.
Prove that the set G = {T|a, bĒR with a # 0} forms a nonabelian group
under composition of functions.
%3D
34. Let H = {TolbeR} (notation as in Exercise 33). Prove that H is an abelian
group under composition of functions.
C. 35. If feS prove that fk=I for some positive integer k, where fk means
fofofo ...of (k times) and I is the identity permutation.
36. Let G = {0, 1, 2, 3, 4, 5, 6, 7} and assume G is a group under an operation *
with these properties:
(i) a + bsa+b for all a, beG;
(ii) a * a = 0 for all a ɛ G.
Write out the operation table for G. [Hint: Exercises 27 and 28 may help.]
11:15 AM
EPIC
O Type here to search
口
Ai
EPIC
50
12/11/2020
Transcribed Image Text:Thomas W. Hungerford - Abstrac x b My Questions | bartleby O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(201.. ... Flash Player will no longer be supported after December 202o. Turn off Learn more of 621 -- A' Read aloud V Draw F Highlight O Erase 205 f d d f| f 31. Let T be a set with at least three elements. Show that the permutation group A(T) (Exercise 12) is nonabelian. 32. Let T be an infinite set and let A(T) be the group of permutations of T (Exercise 12). Let M = {f€ A(T)| f(t) # t for only a finite number of teT}. Prove that M is a group. 33. If a, bER with a # 0, let T:R→R be the function given by Tab(x) = ax + b. Prove that the set G = {T|a, bĒR with a # 0} forms a nonabelian group under composition of functions. %3D 34. Let H = {TolbeR} (notation as in Exercise 33). Prove that H is an abelian group under composition of functions. C. 35. If feS prove that fk=I for some positive integer k, where fk means fofofo ...of (k times) and I is the identity permutation. 36. Let G = {0, 1, 2, 3, 4, 5, 6, 7} and assume G is a group under an operation * with these properties: (i) a + bsa+b for all a, beG; (ii) a * a = 0 for all a ɛ G. Write out the operation table for G. [Hint: Exercises 27 and 28 may help.] 11:15 AM EPIC O Type here to search 口 Ai EPIC 50 12/11/2020
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