Explain why in the following cases the subset X is not a subgroup of the group G: (i) G = Z, X is the set of odd integers (note that the group is additive!); (ii) G = GLn(R), X is the set of matrices A with | det A| ≤ 1; (iii) G = R \ {0}, X = {x ̸= 0, |x| ≤ 1};

Explain why in the following cases the subset X is not a subgroup of the group G: (i) G = Z, X is the set of odd integers (note that the group is additive!); (ii) G = GLn(R), X is the set of matrices A with | det A| ≤ 1; (iii) G = R \ {0}, X = {x ̸= 0, |x| ≤ 1};

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: rphisms : Z2

Q: What is the minimum number of generators needed for Z2 ⨁ Z2 ⨁ Z2?Find a set of generators and…

A: Given: Group, Z2⊕Z2⊕Z2To find: The minimum number of generators and a set of generators and…

Q: Let G :- [0, 1) be the set of real numbers x with 0<x< 1. Define an operation + on G by X* y:= {x+y…

A: Here we check associativity property.

Q: 5. Let G be the multiplicative group {[2], [4], [6], [8]} = Z10-

Q: Show that the Galois group of x 4 + x² + 1 is the same as that of x 6-1 and is of order 2.

A: We have to show that the Galois group of is the same as that of x6 -1 and is of order 2.

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

Q: Prove that a group of order 28 must have an element of order 2.

Q: Prove that every group of order 330 is not simple.

Q: If (G,X) be the group of all real 2 x 2 matrices (b) such that ad - bc #0 with matrix multiplication…

A: Given if G, × be the group of all 2×2 matrices abcd such that ad - dc≠0 with matrix multiplication…

Q: Let G 0 -{(81%) = : a, b ER, a 0 (a) Show that G is a subgroup of SL(2). (b) For each of the…

A: Since you have posted a question with multiple subparts, we provide the solution only to the first…

Q: 7. Consider an element z = the order of r is the least common multiple of the orders of its…

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 (Z4 ⨁ Z12)/<(2, 2)> is isomorphic to one of Z8, Z4 ⨁ Z2, orZ2 ⨁ Z2 ⨁ Z2. Determine…

A: Consider the group elements, Here the order of K is 6. Consider the order of group, The order of G…

Q: O(G) If G is a finite group, than o(G) = Σ- D(N(a)) where this sum is taken over one element of each…

Q: 3. Give the lattice of subgroups of Z₂ X Z₁.

A: Sol:- The group Z2 × Z4 is the direct product of two cyclic groups of order 2 and 4, respectively.…

Q: We will now consider three irreducible representations for the permutation group P(3): E A B I : (1)…

Q: 8. Let (G, *) be a group. Define the center of G by Z(G) := {x € G: x * a = a * x, Va E G}. The set…

Q: Prove that the external direct product of any finite number of groupsis a group.

A: To Prove: The external direct product of any finite number of groups is a group.

Q: 3. Let G, H, K be finitely generated Abelian groups. (i) Show that if G G HH then G= H. (ii) Show…

A: We will solve this by the help of isomorphism

Q: Is U (8) ∼= Z2 ×Z2 ? why? If yes, gives an explicit isomorphism.

Q: Determine the Galois group of x2 -10x + 21 over Q.

A: To determine the Galois group of x2 - 10x + 21 over Q, we need to find the splitting field of the…

Q: (4) Let (G, *) be a group and assume that z, a, b € G. Let also that c = x *a*r¹ and d = x * b*x-¹.…

A: Let be a group and assume that Suppose that Since, and is a group, it follows that…

Q: B (ii), (iii) (i), (ii) (i) (iv) (i), (iv) Suppose f: A B is a non-zero group homomorphism. Which of…

Q: 10 12 14 10 12 14 (b) Calculate dB at t = 2. dt 64 dollars per month (c) Explain what the number you…

Q: Let Zeven represent the set of even integers, {... -6, -4, -2, 0, 2, 4, 6, ...}. Determine if the…

A: (i) Closed: Yes, the set of even integers is closed under addition, meaning that when two even…

Q: (b) Show that for any n ∈ N, the set of all invertible n × n matrices is a group under matrix…

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

Q: 2. a) The operation in a finite group can be specified by writing down a table. Write the table for…

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

Q: f G is a finite group and Z is its centre, then class equation of G is expressible as D(6) 0(N(a))…

A: G is a finite group and Z is its centre, then we have to show that the class equation of G is…

Q: Prove that A5 is the only subgroup of S5 of order 60.

Q: Let G be set of all 2 x 2 matrices in GL2(Z5) of the form (d), (iv) Let N be the subset of all…

Q: An onto grouphomomorphisimf: G H is an isomorpisim if and only if Select one: a. f(1,)=H b.…

A: We have to solve given problem:

Q: set P(S)

A: According to the question, The power set P(S) of a set S is the set of all subsets so, of S,…

Q: Suppose X and Y are rings and that f : X → Y is an isomorphism. Let X2x2 be the set of all 2 × 2…

Q: By applying Fundamental Theorem of group homomorphism, show that the quotient groups GL(n, R)/SL(n,…

A: It is given that, GL(n, ℝ) is the set of general linear group of all n×n matrices and SL(n, ℝ) is…

Q: /rite Z 51 X Z260 as a product of cyclic groups, both in long form, and in Smith normal form.

A: This is abstract algebra problem of classification of finitely generated abelian groups. Please find…

Q: Show that there are no simple groups of order 255 = (3)(5)(17).

Q: If N and K are submodules of M. (i) The kernel of the natural homomorphism N--> M/K is (N+K)/K. (ii)…

Q: 2. Prove that a free group of rank > 1 has trivial center.

A: Given:Prove that a free group of rank>1 has trivial center

Question

Explain why in the following cases the subset X is not a subgroup of the
group G:
(i) G = Z, X is the set of odd integers (note that the group is additive!);
(ii) G = GLn(R), X is the set of matrices A with | det A| ≤ 1;
(iii) G = R \ {0}, X = {x ̸= 0, |x| ≤ 1};
(iv) G = Sn, X is the set of all elements of order ≤ 2;
(v) G = Sn, X is the set of all odd permutations

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps with 1 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

11. Prove that the set of all 2 × 2 matrices with entries from R and de- terminant +1 is a group under matrix multiplication.
The population of a community is known to increase at a rate proportional to the number of people present at time t. The initial population P has doubled in 5 years. Suppose it is known that the population is 11,000 after 3 years. What was the initial population Po? (Round your answer to one decimal place.) Po What will be the population in 10 years? (Round your answer to the nearest person.) persons How fast is the population growing at t = 10? (Round your answer to the nearest person.) persons/year
Build an isomorphism between M2 (R) and P3 (R)
Show that O(n), the set of all orthogonal n × n matrices is a group under matrix multiplication.
(6) Let (G, *) be a group and a € G. Prove that the order of a is equal to the order of a-¹. Prove also that the order of a is equal to the order of b*a*b-¹ for any be G.
Show that any automorphism of GF( pn) acts as the identity onGF( p).
2.10 Let G be the set of all 2x2 matrices (6), where a, bER and a² +b² +0. Show that G forms a group under matrix multiplication.
Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} where a = (123 4) and b = (2 4). Compute (a³b)(a²b). Compute (a³b)(a³). А. a A. а В. a? В. a? С. a3 С. a3 D. b D. b
(7) Define GL2(R) to be the group of invertible 2 x 2 matrices. Show that this group has the structure of a 4-dimensional manifold, conceived as an open subset of R4.
G is a finite group and be an automorphism of G satisfying the condition f(x)= x > x=e for xEG. Show that there can be 36EG for YafG such that a=6"f(6)
2.11 Let G be the set of all 2×2 matrices ), where a and b are nonzero real numbers. Show that G forms a group under matrix multiplication.
True or fulse 17