Exercise 7.9. Let G = Q be the group of rational number with addition. Consider two rational numbers a, b. Show that (a, b) is a cyclic group. (Hint: Show it is contained in a cyclic group.) Deduce that if Q had a finite set of generators, then Q would be cyclic. (We will later see that this is not true.)

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Q: 4. If a is an element of order m in a group G and ak = e, prove that m divides k. %3D

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Q: Show that any group of order 3 is abelian.

A: The solution is given as

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Q: In GL(2, Z,3), det 4. O 35 11

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Q: Compute the left regular representation of the group G = {e,a, b} given by the group table below by…

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Q: 4. Vnite the table for (Z₂ ©) and use it to show that (Z7, © ) is a group. 7 Find the inverse. of 2…

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Q: If G = is a distant group and f: G → G is an automorph, show that f(a) is a generator of G, i.e. G =

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Q: Prove that if x is a group element then |x| = |x-1|.

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Q: 19. Prove that the group (Z/11Z)* is cyclic, and find a generator.

A: We have to prove that the group (Z/11Z)* is cyclic, and find a generator.

Q: Give an example to show that there exists a group with elements a and b such that la] = 0, \b| = ∞,…

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Q: 4. Let G be a group and let a and b be elements of G. Prove that -1 (a*b)¹ -1 =b¹ a

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Q: 1. Build composition table for a dihedral group D4 (Hint: You might need read additional materials…

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Q: Prove that there is no simple group of order 396 = 22 .32 .11.

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Q: 30. Prove the right and left cancellation laws for a group G; that is, show that in the group G, ba…

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Q: Show that (Q*,-) is not cyclic..

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Q: Let G be a group of odd order. Show that for all a E G there exists b E G such that a = b?.

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Q: Prove that there is no simple group of order 300 = 22 . 3 . 52.

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Q: (bläi) äbäi 2.5 3 Jlgull How many abelian groups of order the ?Z180 same as 30O 10 40 30 1.

A: Given: An abelian group of order the Z180

Q: Proposition 15.5.27. Let G be a finite group, and let a e G. Then la| = (a)\. Exercise 15.5.28. Let…

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Q: Let p e Z be a prime number and set Z, = {" e E Q: If ged(n, m) = 1, then p {m. a. Show that Z, 4Q.…

A: Given: ℤp=nm∈ℚ:gcdn,m=1⇒p|m, where p∈ℤ is a prime number.   To show: (d.) Every proper subgroup of…

Q: 1. Let p be a prime number. Find the number of generators of the cyclic group Z where r is an…

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Q: Exercise 14.5.3. Using “switchyard", we proved that S, is generated by the permutations (12) and…

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Q: Prove that (ab)2 = a²b² for all a, b in a group G if and only if G is Abelian.

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Q: 27. Prove or disprove that each of the following groups with addition as defined in Exer- cises 52…

A: Let G = Z2xZ4  i.e G = { (0,0),(0,1),(0,2)(,0,3),(1,0),(1,1),(1,2),(1,3)}  Order of G = 8

Q: If U(14) = ,then U(14) is cyclic group generator by a) 4 b) 11 c) 3 d) None of the above

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Q: Let C be a group and let a be an element of C Prove that = .   Please be clear with theorems ana…

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Q: Suppose H, and H2 subgroups of the group G. Prove hat H1 N Hzis a sub-group of G. are

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Q: 6.2 Which of the following groups are cyclic? a) Z12×Z, b) Z10x Zgs c) Z4xZ2sx Z, d) Z22XZ21 xZ6s

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Q: 5. Write TRUE if the statement is true and FALSE otherwise. Justify your answer. a. All generators…

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Q: Exercise 15.3.6. Fill in the blanks to complete the proof of Proposi- tion 15.3.4

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Q: 52. Let G be a group. Complete the following statement: |Inn(G)| = 1 if and only if

A: Let G be group. Complete the following statement: |InnG|=1 if and only if _________