8. Let S = (12345), T = (13)(45) in S5. Describe the group generated by S and T?

Q: 4.12 write the group table for 2. U(40)/(3)

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Q: 2. Are the groups (R, +) and (R†,') isomorphic? Justify your answer.

A:

Q: 5. Let B = {-1, -i, i, 1}; where i² = multiplication. -1. Show (B,*) is a group where * represents 4

A:

Q: X={1,2,3}. The permutation group S3=

A: Given,            X = { 1,2,3 }

Q: 4. The permutations (e, a, B, v) form a group. If e = (1)(2)(3)(4)(5)(6), a = (1)(2)(35)(46), B=…

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Q: (H,*) is called a of (G,*) if (H,*) is a group.

A:

Q: Show that PSL(2, Z7) = SL(2, Z7)/Z(SL(2, Z7)), which has order 168,is a simple group.

A: Given: PSL2, Z7=SL2, Z7/ZSL2, Z7 has order 168.To show that PSL2, Z7 is a simple group for that we…

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: 7. [10] Let G be a group, with x € G and o (x) = 48. Give the order of the following elements of G:…

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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: Z, x is not a group.

A:

Q: . In the group (Z19 – {ô}, -19), Find the inverse of the following elements. 2 11 (LO

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Q: 6. Is every element of Z = {0,1,2,3, ...} has inverse with respect to addition? 7. Is (Q,.) a group?…

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Q: Show that ( Z,,+,) is a cyclic group generated by 3.

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Q: Is S3 x S3 group (the direct product of symmetric group S3) nilpotent?

A: Given question:   Is S3 x S3 group (the direct product of symmetric group S3) nilpotent?

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: (a) Make addition and multiplication tables for Z/3Z.

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

Q: Prove that the group G with generators x, y, z and relations z' = z?, x² = x², y* = y? has order 1.

A: In order to solve this question we need to make the set of group G by finding x, y and z.

Q: 3. Consider the group (Z,*) where a * b = a + b – 1. Is this group cyclic?

A: 3. Given the group ℤ,* where a*b=a+b-1. Then, 1*x=x*1=x+1-1=x Here 1 serves as the identity for Z.

Q: Let α = (12)(34) and β = (24). Show that the group generated by αand β is isomorphic to D4.

A: It is given that α=1234 and β=24. It is required to show that the group generated by α and β is…

Q: 1. Consider H= (i) and K= (-1) the subgroups of Qg the quaternion group. 4) Explain why K is normal…

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Q: 3. Show that G = {a+bv2 |a,b e Q} is a group under the usual addition.

A: 3. Let G is said to be group under binary operation (.). It should satisfy the following properties.…

Q: Let (G, *) be a group. Prove the following. (a) For each y = G, (y-¹)-¹ = y. (b) For x, y = G,…

A:

Q: Find all the producers and subgroups of the (Z10, +) group.

A: NOTE: A group has subgroups but not producers. Given group is ℤ10 , ⊕10 because binary operation in…

Q: 13335 2.2. SYMMETRIC AND ALTERNATING GROUPS Exercise 52. Compute the orders of the following…

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Q: 46. Determine whether (Z, - {0}, 6 ) is it a group or not? Explain your answer?

A:

Q: 2. Are the groups Z/2Z x Z/12Z and Z/4Z x Z/6Z isomorphic? Why or why not?

A: Here we have to show that given groups are isomorphic

Q: Decide whether (Z, -) forms a group where : Z xZ Z (a) is the usual operation of subtraction, i.e.…

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: 16. Give a specific example of some group G and elements g, h = G where (gh)" # gh".

A:

Q: 19. 20. 21. 22. 23. 24. 25. Consider the groups U(8) and Z4 c) d) e) f) Let H₁, H₂ and H, be abelian…

A: As per our guidelines we do question no. 19 with its first three subparts and rest questions can be…

Q: 9. a) find two integers x,y such that 30x+101y=1. (hint: gcd(30,101)=1).     b) find the inverse of…

A: Given: We have given numbers 30 and 101 We want to find integers x and y such that 30x+101y=1