Please if you someone could explain part B in an easy, concise way for me. Greatly appreciated. 6. Let G be a group, and a an element of G. The \emph{centralizer} of a in G isC(a)= {gG: ga = ag}.a. Show that C(a) is a subgroup of G.b. If G = D4 and a = B is horizontal reflection (see p.~58), find C(3). 58111/Semigroups, Monoids, and Groups► (1.6) EXAMPLE: D₁, the Symmetries of a Square.Although the study of groups usually seems quite abstract at first, muchof the motivation for groups comes from geometric problems, especially problems concerning symmetries of geometric figures. By a symmetry of a square.we mean, roughly speaking, a rigid motion of the square in 3-space so thatthe square covers the same region at the end of the motion as it did atthe start. More precisely, if we take axes for 3-space with the origin at thecenter of the square, then a symmetry is a linear transformation of 3-space.that carries the square onto itself. Such a linear transformation must be arotation, or a reflection about some line, or a composition of rotations andreflections. We will assume all rotations below to be in the counterclockwisedirection.(1.7) DEFINITION. Let D, denote the set of symmetries of a square.More generally, for n 23, let Dn denote the set of symmetries of a regularplane n-gon (i.e., a polygon in the plane with n equal sides and each exteriorangle equal to 360°/n).Figure 1.1For the square in Figure 1.1, let a denote the symmetry given by rotat-ing the square (about its center) by 90° and let 3 denote the symmetry givenby reflection about the line h. Again, composition of mappings is an obviousoperation to be considered on D4 and again we will write the operation as"multiplication." The set D₁ then has the following eight elements:(= the identity mappinga = rotation by 90°a²rotation by 180°a³ rotation by 270°3= reflection about the line haß= reflection about the diagonal BDa²3 reflection about the line va³3= reflection about the diagonal AC.1/ExamplesIt is not too hard to determine the following multiplication table for D4-(1.8) TABLE(a0²03Baß0²6a³6€9²a³Baåa²8a³8a0²9³€0³3Baßa²30²a²QªCaa²3a³3BQ³a³C0²aßa²8a³3Baß28²8265aßa²8aßa²3a³8a³8 3 aßa²BB9³€a0²939²qªCaFigure 1.2a³BaBB59a9²03One may check that D4 with composition satisfies the same four prop-erties as S3 did, hence that D₁ is a group. It is called the dihedral groupof order 8. We note that each reflection is its own inverse, as is rotation by180°; rotation by 90° and rotation by 270° are inverse to each other. We willstudy Dn, the dihedral group of order 2n, further in §8. ◄3Note that if we label the four vertices of our square 1, 2, 3, and 4,as in Figure 1.2, then we may represent a symmetry of the square by anelement of S4 by considering the final position of each vertex after applyingthe symmetry. For example, (counterclockwise) rotation by 90° will takevertex 1 to position 2, vertex 2 to position 3, etc. Hence we may representa by the permutation(23³1)The reader may check that underthis correspondence the product of two symmetries will be represented bythe product of the two corresponding permutations. It is often useful torepresent the elements of a group by permutations on some set and, as wewill see, this is always possible.

Let G be a group and a an element of G Ca=g∈G: ga=ag

6. Let G be a group, and a an element of G. The \emph{centralizer} of a in G is C(a) = {g G: ga = ag}. a. Show that C(a) is a subgroup of G. b. If G = D₁ and a = ß is horizontal reflection (see p.~58), find C(B).

6. Let G be a group, and a an element of G. The \emph{centralizer} of a in G is C(a) = {g G: ga = ag}. a. Show that C(a) is a subgroup of G. b. If G = D₁ and a = ß is horizontal reflection (see p.~58), find C(B).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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