(a) Describe the symmetries of the Figure below. Assume that o is the centre of rectangleabcd.dab(b) Give the Cayley table for the symmetries of Question 1(a).Given mappings a: A→B,B: B → C, and y: C → D.(c) Prove associativity for these mappings, and then show that every invertible mapping hasa unique inverse.

(a.) To describe: The symmetries of the given figure, assuming that o is the center of rectangle…

Answered: (a) Describe the symmetries of the…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 5E

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Let A be a given nonempty set. As noted in Example 2 of section 3.1, S(A) is a group with respect to mapping composition. For a fixed element a In A, let Ha denote the set of all fS(A) such that f(a)=a.Prove that Ha is a subgroup of S(A). From Example 2 of section 3.1: Set A is a one to one mapping from A onto A and S(A) denotes the set of all permutations on A. S(A) is closed with respect to binary operation of mapping composition. The identity mapping I(A) in S(A), fIA=f=IAf for all fS(A), and also that each fS(A) has an inverse in S(A). Thus we conclude that S(A) is a group with respect to composition of mapping.
The group of all rotations in space that preserve a given cube has order: (u) 6.(b) 12.(c) 8.(d) 24.(e) 48.
Consider the example where G is the symmetries of the square, and H is the rotations of thesquare. Then the two cosets of H are { elt[1], elt[2], elt[3], elt[4] } = H = H xid, and{ elt [5], elt [6], elt [7], elt [8] ] = H • elt [5]. A set of coset representatives is{ elt[1], elt[5] } and the reader can verify that each element is uniquely expressible as aproduct of an element of H and a coset representative.
Theorem [4.1.9] The set S = {T, (U) | U is open in X} U {T2 (V)|V is open in Y} is a subbasis for the product topology on X × Y. -1 Proof : H.W
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Let G ={[1 0 0 1] ,[−1 0 0 1] ,[1 0 0 −1], [−1 0 0 −1]} . Is G ∼= K4? If yes, givean explicit isomorphism?
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Suppose T : Rn → Rn is a linear transformation. Prove that T is an isometry if and only ifT(v) · T(w) = v · w. Recall that an isometry is a bijection that preserves distanceNote: When proving that if T(v)·T(w) = v·w then T is an isometry, make sure you verify that T is a bijection.
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