Let a,ß ESg(Symmetric group) where a=(1,8,5,7)(2,4) and B= (1,3,2,5,8,4,7,6). Compute aß-

Q: Construct the group table for (Z,, + ,).

A: We have to construct the group table for (Z5,+5). Where Z5 is modulo 5 group of integers. So to…

Q: In (Z12, +12) , H, = {0,3,6,9} and H2 = {0,3} are tow subgroups of the group (Z12, +12), but (H, U…

A: We have given   H1=0,3,6,9 and H2=0,3 are two subgroups of the group Z12, +12. We can see here…

Q: a. Show that (Q\{0}, * ) is an abelian (commutative) group where * is defined as a ·b a * b = .

A: To show this we have to show that this holds closure, associative, identity, inverse and commutative…

Q: Give the lattice of subgroups of (Z2,), of (S,,0), and of (Z(20) ,)

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Q: List two examples of nontrivial proper subgroups of the indicated group. a) Z18 b)U(18)

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Q: (Z, +) is a group and infinite group

A: Let a binary operation '*' defined on a set G, then it forms a group (G,*) if it holds the following…

Q: If a1, a2,...,an belongs to group .what is inverse of a1a2...an

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Q: Verify that (Z, O) is an infinite group, where Z is the set of integers and the binary operator O is…

A: According to the given information, it is required to verify that:

Q: 7. Show that 4 is a subgroup of S,

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Q: Prove that {(1), (1, 2)(3, 4), (1, 3)(2,4), (1, 4)(2,3)} is a subgroup of S4.

A: We use the Cayley table.

Q: The group generated by the cycle (1,2) is a normal subgroup of the symmetric group S3. True or…

A: Given, the symmetric group S3={I, (12),(23),(13),(123),(132)}. The group generated by the cycle (12)…

Q: Explain why S8 contains subgroups isomorphic to Z15, U(16), and D8.

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Q: Let S, be the set of all one-one and onto mappings on {1,2,3} and Z, = (0, 1,2, 3, 4}. Then, show…

A: Let G be a group and ∗ be an operation defined in G. Then (G, ∗) is group if it satisfies following…

Q: Is the set Z a group under the operation a * b = a + b – 1? Justify your answer.

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Q: Show that S5 does not contain a subgroup of order 40 or 30.

A: Let’s assume that the H is a subgroup of S5. So,

Q: 10. Prove that any group of order 40, 45, 63, 84, 135, 140, 165, 175, 176, 189, 195, 200 is not…

A: Given that,   (a) Group of order 40. Let G=40,  i.e, G=40=23.5, Since, 5-Sylow subgroup, i.e…

Q: Verify that (Z, O) is an infinite group, where Z is the set of integers and the binary operator O is…

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Q: Suppose that H is a subgroup of Z under addition and that H contains250 and 350. What are the…

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Q: In a group G,let a,b and ab have order 2.show that ab=ba

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Q: Z, x is not a group.

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Q: 3. Define Lie group.

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

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Q: Compute the center of generalized linear group for n=4

A: To find - Compute the center of generalized linear group for n=4

Q: Is it possible to find a group operation e on a set with 0 elements? With 1 element? Explain why or…

A: The question is :: is there possible to find a group operation on a set of 0  element? Or with 1…

Q: The group Cs3,0) is normal group solvable ?

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Q: Every subset of every group is a subgroup under the induced operation. True or False then why

A: True or FalseEvery subset of every group is a subgroup under the induced operation.

Q: Suppose that <a>, <b> and <c> are cyclic groups of orders 6, 8, and 20,…

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Q: {a3 }, {a2 }, {a5 }, {a4 }   Which among is not a subgroup of a cyclic group of order 12?

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Q: G={(1),(174)(258),(147)(285),(174)(258)(36),(147)(285)(36),(36)} be a group which acts on the set S…

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Q: The set A := {(1), (1 2 3), (2 3 4)} forms a subgroup of the permutation group (G,.). O True O False

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Q: Let Ø:Z50→Z15 be a group homomorphism with Ø(x)=7x. Then, Ker(Ø)= * O {0, 10, 20, 30, 40} None of…

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Q: (i). There is a simple group of order 2021.

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Q: QUESTION 3 Construct the group table for (U(9), ).

A: 3 We have to construct the group table for U9,⋅9. First of all we will write the element of U9,…

Q: 8. Give an example of a group G where the set of all elements that are their own inverses does NOT…

A: Let, G,. is a group. Let, G={1,7,17,2,12,3,13} Let, H be a subgroup of G where H={1,7,17,2,12}

Q: The group (Z6,6) contains only 4 subgroups

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Q: Show that if aEG, where G is a group and |a| = n then : %3D a' = a' if and only if n divides. -j

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Q: Is the set Z a group under the operation a * b = a – b + ab? Justify your answer.

A: Check the associative property. Take a = 2, b = 3 and c =4. (a*b)*c = (2*3)*4              =…

Q: Let a,B ES ( Symmetric group) where a = (1,8,5,7)(2,4) and B=(1,3,2,5,8,4,7,6)- Compute aB. Attach…

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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: determine whether the binary  operation * defined by a*b=ab gives group structure of Z. if it is not…

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Q: The group (Z6,+6) contains only 4 subgroups

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Q: Check whether a group of order 156 is simple or not.

A: Definition: A group G is said to be simple if the only normal subgroups of G are the trivial group e…

Q: Theorem 2. Let G, and G, be two groups. Let G = G,x G2 H = {(a,e,)\a e G} = G, x{e,} %3D and H, =…

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Q: Let alpha, beta in S8 ( Symmetric group) where alpha=(1,8,5,7)(2,4) and beta=(1,3,2,5,8,4,7,6).…

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Q: Show that a group of order 12 cannot have nine elements of order 2.

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

Q: Let a,b be elements of S6 (symmetric group) where a=(1,2)(4,5) and b=(1,6,5,3,2).verify that…

A: Given: Let a,b be elements of S6 (Symmetric group) where a=(1,2)(4,5) and b=(1,6,5,3,2).

Q: G be the external direct product of groups G, G2.. H, = {4,e2.e*, e..e,x, e G,}

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Q: The group (Z, t6) contains only 4 subgroups

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Q: Verify that (ℤ, ⨀) is an infinite group, where ℤ is the set of integers and the binary operator ⨀ is…

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Q: Let G = Zp × Zp. Is this group cyclic? As you know any cyclic group can be generated by one element.…

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Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.