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.)

Q: Exercise 15.5.33. (a) Find all cyclic subgroups of the symmetry group of the square (i.e. D₁) by…

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

A: We have to prove that there is no simple group of order 396=22⋅32⋅11. We will prove this by the…

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..

A: We use the basic definitions of cyclic groups to solve this problem. The detailed solution is…

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?.

A: Consider the given information, Let G be a group of odd order then, |G|=2k+1 where k belongs to…

Q: Problem 8. Suppose that G = (a) is an infinite cyclic group. a) Prove that every non-identity…

A: Suppose that G = <a> is an infinite cyclic group.a) Prove that every non-identity element of G…

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: 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: 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: 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: 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 _________