Q4: Consider the two group (Z, +) and (R- {0}, ), defined as follow if n EZ, f(n) ={1 if nE Z, %3D Prove that f is a homomorphism and find ker(f).

Q: Let G = {x ∈ R : x 6= −1} . Define △ on G by x△y = x + y + xy   Prove that (G, △) is an abelian…

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Q: Explicitly construct the Galois group for r- 4r² + 2 over Q. To which group is this isomorphic?

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Q: 2- Let (C\{0},.) be the group of non-zero -complex number and let H = { 1,-1, i,-i} prove that (H,.)…

A: To Determine: prove that H,. is a subgroup of a group of non zero complex number under…

Q: Let G be a group with the property that for any x, y, z in the group,xy = zx implies y = z. Prove…

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Q: Consider the square X = [-1,1]2 = {(x, y)|x > -1, y < 1} and 0 = (0,0). Show that the fundamental…

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Q: Q1) Consider the group Z10X S5. Let g = (2, (345)) € Z10X S5. Find o(g). T LOV

A: as per our company guideline we are supposed to answer only one qs kindly post remaining qs in next…

Q: Define the mapping 7: R²→R by π((x,y))=x. (Note that R is a group under addition with identity 0).…

A: Here we use the definitions of group homomorphism and the kernel of it . Which are given in solution…

Q: Q3:(A) Prove that every group of order 15 is decomposable and normal. (B) Show that (H,.) is a…

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Q: be the operation on Z defined by a*b = a+b for all a,beZ. Justify the following questions. 4 Let (1)…

A: Here * be the operation on ℤ defined by a*b=a+b4 for all a, b∈ℤ. We have to justify the followings:…

Q: V2n 5) Let G be a group such that |G| = (e" xd,)!, and |H|= (n– 1), where H a %3D Subgroup of G,…

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Q: Let G = (Z;, x,) be a group then the order of the subgroup of G generated by 2 is О а. 6 O b. 3 О с.…

A: We have to find order of subgroup of G generated by 2.

Q: Let G be a group and let g, h ∈ G. Show that | gh | = | hg |. Remember that | a | denotes the order…

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Q: et G be a group with order n, with n > 2. Prove that G has an element of prime order.

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Q: Q5. Let A and B be two groups. Let 0: A x B → B defined by 0(a, b) = b Is 0 isomorphism? Find…

A: To check whether a function θ is isomorphism, it is required to check  θ is homorphism θ is one-one…

Q: 9. Classify the following groups in the sense of the Fundamental Theorem of Finitely Gener- ated…

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Q: In the following problems, let G be an abelian group and prove that the set H described is a…

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Q: I. Provide a two-column proof to the following statements. 1. Let G be a group with the property…

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Q: 16* Find an explicit epimorphism from S5 onto a group of order 2

A: To construct an explicit homomorphism from S5 (the symmetric group on 5 symbols) which is onto the…

Q: implies y = z. Prove that G is Abelian. Prove that a group G is Abelian if and only if (gh)-1 = g¬…

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Q: %3D Let x belong to a group. If x² +e while x° = e, prove that about the order of r?

A: Given that x2≠e and x6=e To prove that x4≠e and x5≠e Suppose that x4=e  also x6=e therefore…

Q: Exercise 1) Consider the group (S3, 0) and H= {e, fs). Prove that HS3. 2) Consider the group (Z.)…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: 1+2n Prove that if (Q-(0},) is a group, and H = a n, m e Z} 1+2m is a subset of Q-{0}, then prove…

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Q: If f (x) is a cubic irreducible polynomial over Z3, prove that either xor 2x is a generator for the…

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Q: True or false? The group S3 under function composition ◦ is not a cyclic group

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Q: Q3:(A) Prove that every group of order 15 is decomposable and normal. (B) Show that (H,.) is a…

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Q: How do we describe all the elements in the cyclic subgroup of GL(2, R) generated by the matrix 1 ? 0…

A: For the given statement

Q: Consider the group G (x E R]x 1} under the binary operation : ** y = xy-x-y +2 If x E G, then x =…

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Q: Let (G,*) be a group such that x² = e for all x E G. Show that (G,*) is abelian. (Here x² means x *…

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Q: Q3: (A) Prove that 1. There is no simple group of order 200.

A: Simple group of order 200

Q: 2. Let G = (1, 0). Decide if G is a group with respect to the operation * defined as follows: x * Y…

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Q: 2. Consider the group (R, +) and the subgroup H = (2nx |n€ Z}. %3! [cos(0) - sin(@) sin(0) cos(0)…

A: Kindly keep a note in step 1 stating that as per our guidelines we are supposed to answer only one…

Q: 2. Consider the groups (R, +) and (R x R, +). Define the map : RX (RX R) → Rx R defined by r(x, y) =…

A: Given: The group is, (ℝ, +) and (ℝ×ℝ, +) The map is, ℝ×(ℝ×ℝ)→ℝ×ℝ To find: a) The defined map is an…

Q: . Deduce from 1 that V x Z2 is a group where V = {e, a, b, c} is the Klein-4 group. (a) Give its…

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Q: Let G = Z, be the cyclic group of order n, and let S c Z, \ {0}, such that S = -S, \S| = 3 and (S) =…

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Q: 5. Consider the group (R+, 0). Prove that the function F: R -R given by: F (x.y) = (x +y.r-y) is a…

A: F: R2 → R2F (x, y) = (x +y, x-y)Let (x1, y1) ,  (x2, y2) = R2 (x1, y1) +  (x2, y2) =  (x1+ x2, y1+…

Q: Q2 : Find the left regular representation of the group Z5 and express the group element in the…

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Q: 2. Consider the groups (R, +) and (Rx R, +). Define the map: RX (RXR) → Rx R defined by r(x, y) =…

A: Note:  Since we can solve at most one problem at a time we have solved the first problem that you…

Q: How do we describe all the elements in the cyclic subgroup of GL(2, R) generated by the matrix 1 1 0…

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Q: Let S = R\{-1}. Define * on S by a * b = a+b+ ab. Prove that (S, *) is an abelian group.

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Q: If I is an ideal of R, then by definition, (I, +) is an abelian group. Consequently, it has an…

A: Since I is ideal , therefore I is definitely a subset of the ring R.

Q: How do we describe all the elements in the cyclic subgroup of GL(2, R) generated by the matrix 1 ? 1…

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Q: 5. Prove that the cyclic group Z/15Z is isomorphic to the product group Z/3Z × Z/5Z.

A: Definitions: Isomorphism: A mapping between two sets is called an isomorphism if it is one-to-one,…

Q: Exercise 1) Consider the group (S3, 0) and H= {e, f3}. Prove that HS3. 2) Consider the group (Z¸ +)…

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Q: Let Hand K be subgroups of an Abelian group. If |H| that HN Kis cyclic. Does your proof generalize…

A: This question is related to group theory. Solution is given as

Q: Let Dn (n ≥ 3) be the dihedral group of order 2n. (i) Show that D10 D5 x Z2 by constructing an…

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Q: Let G = {a + b/2|a, b € Z}. Show that G is a group under ordinary addition.

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Q: Let GL(2,R) be the general linear group of 2 by 2 matrices. 1 x is a) Show that o: R GL (2,R) where…

A: We first prove this is a homomorphism.

Q: 25. Prove that R* x R is a group under the operation defined by (a, b) * (c, d) = (ac, be + d).

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Q: 1+2n 1- Prove that if (Q – {0},') is a group, and H = { a n, m e Z} 1+2m is a subset of Q – {0},…

A: NOTE:  We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: a. Show that (Q\{0}, + ) is an abelian (commutative) group where is defined as a•b= ab b. Find all…

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Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.