Let G be group. Assume that Set H₁ = and H₂ =

Q: Prove that the chromatic number of G is 4. Ie: x(G) = 4.

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Q: Let C2 = {±1} be a subgroup of R". Prove that R = R+ x C2 where R is the set of real numbers.

A: Please check step 2 and 3 for the solution.!

Q: SIZ): Set of all fiZ-→Z which are both one-one S(Z): on ti and S(Z) is a group under composition of…

A: Given, S(ℤ)=f|f:ℤ→ℤ, 1-1, onto is a group under composition of function. And H=fn:ℤ→ℤ|fn(x)=n+x.…

Q: Q,+ ,+,*) is not a proper subgroup of R+ under multiplication? T or F

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Q: 1.e. Consider the dihedral group (D3, ◦), ie. the symmetry group of an equilateral triangle.…

A: Symmetry group of an equilateral triangle have subgroup of order 2 and it is isomorphic to Z2

Q: 7.) Name 3 different (single) transformations that would map this figure to itself. -2 3 A. B. -2-…

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Q: Show that the set A = {fm,b : R → R | m # 0 and fm,b(æ) group under composition of functions. (The…

A: To Show- Show that the set A = fm,b : R → R | m ≠ 0 and fm,bx = mx + b, m, b ∈ R forms a group under…

Q: In the frieze group F7, show that zxz = x-1.

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Q: Let ā = ( – 5, — 4, 5) and - 4, 5) Find the projection proja b 6 and 7 = (3, 1, 0). =< of b onto a.…

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Q: Let à = ā ( – 5, — 4, 5) and b Find the projection of b onto a. proja b = (3, 1, 0). V

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Q: Question 3: a. Let (H,+) ≤ (R, +), let K = {2h : h E H}. Prove that (K,x) ≤ (R*,x). (Clarification:…

A: We shall use subgroup test to solve this problem

Q: d) G=(Q-(0}, ); H=Q+ e) G=(Zg, ); H=(0,2,4} f) G=the set of 2-tuples of real numbers (a,b) under…

A: In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of…

Q: Let G be a group. a) Define the center Z(G) b) Show that Z(G) is a subgroup of G c) Show that Z(G)…

A: WE HAVE GROUP G  THEN WE HAVE TO DEFINE CENTRE OF G AND SHOW THAT IT IS SUBGROUP AND NORMAL SUBGROUP…

Q: prove that v<e+1 must contin a cycle

A: Let G be a graph with number of vertices, v and number edges, e. To prove that if v&lt;e+1, then G…

Q: Let à = (1, 1, 4) and b = (1, 2, - 5). of b onto a. - Find the projection proj- 6 = b =< -

A: For the given vectors the projection of b on a is given by

Q: Let C be a group and let a be an element of C Prove that = .   Please be clear with theorems ana…

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Q: 5. Prove (using the subgroup condition rather than using the axioms) that the set a b H = {( 6 ) :…

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Q: Claim that = (0.0.0) 2 a) show √ E OF √₂, √₂. b) Show OF J₁ () show J₁ s={√₁₁ √√₂, √√₂} S= √√₂₁ are…

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Q: What is the domain of the graph above? O A) [-2, 0) Ов [-5, — 2)U (-1, o) C) [-2, 3] O D) [-5, – 2)…

A: Solution

Q: Show that group Un (n th unit root) and group Zn are isomorphic.

A: There are n elements in the group (Zn,+). There are n elements in the group (Un,×). There are (n!/2)…

Q: 2) Let G = V4.Show which of the following subsets of G is a subgroup of it? a) H = {e, a} b) K = {e,…

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