2.11. The group S3 consists of the following six distinct elements e, σ, σ’, τ, στ, σετ, where e is the identity element and multiplication is performed using the rules 0³ = e, 7² = 1, TO = = 0²T. Compute the following values in the group S3: (a) To² (c) (OT) (OT) (b) τ(στ) Is S3 a commutative group? (d) (OT)(0²T).

Q: Find the order of the following elements in the given groups: a. The order of sr15 in D50 is o…

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Q: Let G = (Z;, x,) be a group then the inverse of the elements 2, 3 and 6 are O a. 3, 4 and 6 O b. 1,…

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

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

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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: Consider the factor group G = R* /Q*. Determine which of the following elements in G are equal. 1Qx,…

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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: 3.8.3 If G is a matrix group with identity component H, show that AHA CH for each matrix AEG.

A: To Determine :-  If G is a matrix group with identity component H , show that AHA-1 ⊆ H for each…

Q: Question 3 [6 points each]. What is the inverse of each of the following elements: 1. 3 in (Z100,…

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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: السؤال 17 2.5 نقطة )نقاط( For the group D3, the value of U2 µ3 is Po

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Q: X={1,2,3,4} .The permutation group S4=

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Q: Let Zodd represent the set of odd integers, {... -5, -3, -1, 1, 3, 5, ...}. Determine if the system…

A: The given set is Zodd=. . . . . . -5, -3, -1, 1, 3, 5, . . . . . . . We have to determine  (i) Zodd,…

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: Let G = <a, b | a6 = b3 = e, b-1ab = a3>. How many elements doesG have? To what familiar group…

A: Given set is G = <a, b | a^6 = b^3 = e, b^-1ab = a^3>.

Q: Let G = <a, b = a3 =e, b2 =e, aba-1b-1=e>. To what familiar group is G isomorphic?

A: As per the question we are given a group G such that :G = {a, b, e}, where : a3 = e, b2 = e, aba-1 =…

Q: Consider the group D4

A: Given: Group D4=a,b=e=1,a,a2,a3,b,ab,a2b,a3b and a=1 2 3 4, b= 2, 4 To find : The value of…

Q: Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} where a = (1 2 3 4) and b =…

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Q: How many subgroups of the group G = (Z30, +30)? а. 8 O b. 6 С. 5 O d. 7 O e. 4

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Q: ACR group A = {x+y: −1 < x≤0, 2 < y ≤ 3}. Define group A's infinum (inf A).

A: As  -1 &lt; x And  2 &lt; y Then   -1+2 &lt; x+y I. e. 1 &lt; x+y

Q: Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} %3D where a = (1 23 4) and b…

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Q: 2 3 4 1 4 3 2 3 4 1) 1 2 2. Let X = {1,2, 3, 4}. Consider the following elements of P(X): a = (; 2 3…

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Q: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({a +bv2 : a,…

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Q: Find S4/N elements and show that they can be combined into a group. N={e,(23),(234),(243),(12)(34),…

A: We know,      | S4 | = 4! = 24     | N | = 12   So, | S4/N | = | S4 |/| N |                      =…

Q: Find the elements in the alternating groups A3, and A4, respectively.

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Q: 34. Suppose a and b are group elements such that |a| = 2, b ‡ e, and aba = b². Determine [b].

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Q: The following is a Cayley table for a group G. 2* 5*4 = 1 2 3 5 2 3 4 3 4 2 3 5 1 4 2 3 4 1 2 4 1.…

A: Cayley table for a group G is given as, The objective is to find 2*5*4 Since, G is a group. Hence,…

Q: Ex. 29 Give the Cayley table for the group Zi2 under multiplication modulo 12.

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Q: Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} %3D where a = (1 2 3 4) and b…

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Q: Q17 : If a and b are elements in a group G, then the order of ab is the same as the order of ba.…

A: Let us take a non abelian group then their elements will never commute  that is a*b not equal to b*a…

Q: Find the order of the element (2, 3) in the direct product group Z4 × 28. Compute the exponent and…

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Q: Exercise 15.3.10. (a) Suppose a, b € C*, where a = 4 + 3i and b = 5 - 12i. What is (ab)-¹2 What is…

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Q: List all the elements of the cyclic subgroup of U(15) generated by 8. 2. Which of the following…

A: We have to find the all elements of cyclic subgroup of U(15) generated by 8.

Q: Find all the generators of the following cyclic groups: (Z/6Z,+), ((Z/5Z)*, ·), (2Z, +), ((Z/11Z)*,…

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Q: 3 In the group (S5, o), (142) o (123)= ....

A: The above problem can be solved as follows:

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: How to complete the table using the binary operation as a group? Let, S={f,g,h,j}, and let the…

A: Given the set