Let G = (Z,, +6) is an Abelian group then how many self - invertible elements in G? (A) 1 (B) 2 (C) 3 (D) 4

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: 6. Apply Burnside's formula to compute the number of orbits for the cyclic group G = {(1,5) o (2, 4,…

A: Solve

Q: 1. Let G be an abelian group with the identity element e. If H = {x²|x € G} and K = {x € G|x² = e},…

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Q: Find two elements of maximum order in the group G = Z100 Z4 O Z2. How many such elements are there?…

A: As Z100 has no element of order 100 otherwise it will be cyclic which is not true. So, we consider…

Q: Prove that a group of order 175 is Abelian.

A: Let G be a group of order 175 We first try to rewrite 175 as prime factorization as follows: 175 =…

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

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Q: (1) Z/12Z (2) (Zx 끄)/(6Zx 14Z) (3) (Z4 × Z4)/((2, 3)) (4) (Z4 x Z10)/((2, 4))

A:

Q: (c) Prove that if G is a (not necessarily abelian) group, a, b e G, and a² = b² = (ab)² = e, then ab…

A: Use property of group and solve it.

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: 6. Let G be a group with the following property: "If a, b, and c belong to G and ab=ca, then b = c."…

A: For a group to be abelian it should follow that ab=ba for all a,b belonging to G. Now we will…

Q: nilpotent

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Q: Consider Z2 = {¯0, ¯1} and check whether the general linear group GL(2,Z2) of 2 × 2 matrices over Z2…

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Q: (a) Show that a group G is abelian, if (ab)² = a²b², for a, b € G[C.

A:

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: Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)-1 = a-'b-1 %3D

A: Given: The statement is, let a and b be elements of a group G. Prove that G is abelian if and only…

Q: QUESTION 7 Show that the special linear group, SL(2, R) is non -Abelian.

A:

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

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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: Let G={-1,0,1}. Verify whether G forms an Abelian group under addition.

A: G  is a group under '+' if  (i) a , b E G  -----> a + b E G     (ii) a E G called the identity…

Q: Describe all the elements in the cyclic subgroup of GL(2,R) generated by the given 2 × 2 matrix. -1

A: Let A=0-1-10∈GL2, R Then the cyclic subgroup generated by A denoted by A=An| n∈Z

Q: prove: let g be a group, if g is abelian then (ab)^2 = (a^2)(b^2)

A: Given g is an abelian group.

Q: (1) Z/12Z

A:

Q: 6. ! Write down the relation matrix of the abelian group G specified as follows. G - {x, y, z, w |…

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Q: Prove that a group G is abelian if and only if (ab)-1 = a¬b¬1 va,bEG

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Q: If (G, ) is a group with a = a for all a in G then G is %D abelian

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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: 2. Let (G. .) be a group such that a.a = e for all a EG. Show that G is an abelian group.

A: Definition of abelian group : Suppose <G, .> is a group then G is an abelian if and only if…

Q: Consider the group G = {x € R]x # 1} under the binary operation : *• y = xy – x-y +2 The identity…

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Q: 52

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Q: (2) (Z x Z)/(6Z × 14Z)

A:

Q: Let G = (Z6, +6) is an Abelian group then how many self - invertible elements in G? 1.a O 4 .b O 3.c…

A: Let G = (Z6 , +6) is an abelian group. We know Z6 = {0,1,2,3,4,5} An element is said to be self…

Q: Determine the class equation for non-Abelian groups of orders 39and 55.

A: We have to determine the class equation for non-Abelian groups of orders 39 and 55.

Q: (7) Define GL2 (R) to be the group of invertible 2 x 2 matric manifold, cc this group has the…

A: Define GL2R  to be the group of invertible 2×2 matrices.  To prove that this group has the structure…

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

A: A simple group is a nontrivial group whose only normal subgroups are the trivial group and the group…

Q: 2) Prove that Zm × Zn is a cyclic group if and only if gcd(m, n) cyclic group Z; x Z4. = 1. Find all…

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Q: Show that the set S = (1, i, - 1, -0 is an abelian group with respect to the multiplication

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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 1 0…

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Q: 1. Show that the group (a, b, c|bª = a, c¬1 = b) is abelian. %3D

A: The objective is to show that the given group is abelian. Given group is: a,b,c|b4=a,c-1=b

Q: prove that a group G of order p^2, where p is a prime, is abelian.

A: Suppose, G is a group of order p2 where p is prime.

Q: Let G be a group. V a, b, c d and x in G, if axb cxd then ab = cd then G is necessa: Abelian Of…

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Q: Construct the Cayley table for (Zo) ,c), and verify that this is an Abelian group.

A: 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 0 2 3 4 5 6 7 8 0 1 3 4 5 6 7 8 0 1 2 4 5 6 7 8 0 1 2 3…

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

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

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Q: Let (G,*) and (H,*) be finite abelian groups. If G x G = H x H then G=H. Show that they are…

A: Given that, G×G=H×H⇒G=H Since G,* and H,* are both finite abelian groups we get,…

Q: Prove that a group G is Abelian if and only if (ab)-1 = a-1b-1 forall a and b in G.

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

Q: Let G be a group and suppose that (ab)2 = a²b² for all a and b in G. Prove that G is an abelian…

A:

Q: Let G be a group. V a, b, c d and x in G, if axb = cxd then ab = cd then G is necessarily:…

A: The answer is given as follows :

Q: 2. Show that the group GL(2,R) is non-Abelian, by exhibiting a pair of matrices A and B in GL(2, R)…

A: Take the matrices from GL(2,ℝ).

Q: Prove that the symmetric group (S₂, 0) is abelian.

A:

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.