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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
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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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 is C(a)= {g G: 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).
Transcribed Image Text: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 = D4 and a = B is horizontal reflection (see p.~58), find C(3).
58 111/Semigroups, Monoids, and Groups ► (1.6) EXAMPLE: D₁, the Symmetries of a Square. Although the study of groups usually seems quite abstract at first, much of the motivation for groups comes from geometric problems, especially prob lems 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 that the square covers the same region at the end of the motion as it did at the start. More precisely, if we take axes for 3-space with the origin at the center 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 a rotation, or a reflection about some line, or a composition of rotations and reflections. We will assume all rotations below to be in the counterclockwise direction. (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 regular plane n-gon (i.e., a polygon in the plane with n equal sides and each exterior angle equal to 360°/n). Figure 1.1 For 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 given by reflection about the line h. Again, composition of mappings is an obvious operation 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 mapping a = rotation by 90° a² rotation by 180° a³ rotation by 270° 3= reflection about the line h aß= reflection about the diagonal BD a²3 reflection about the line v a³3= reflection about the diagonal AC. 1/Examples It is not too hard to determine the following multiplication table for D4- (1.8) TABLE ( a 0² 03 B aß 0²6 a³6 € 9² a³ B aå a²8 a³8 a 0² 9³ € 0³3 B aß a²3 0² a² Qª C a a²3 a³3 B Q³ a³ C 0² aß a²8 a³3 B aß 28²8265 aß a²8 aß a²3 a³8 a³8 3 aß a²B B 9³ € a 0² 93 9² qª C a Figure 1.2 a³B aB B 59 a 9² 03 One 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 group of order 8. We note that each reflection is its own inverse, as is rotation by 180°; rotation by 90° and rotation by 270° are inverse to each other. We will study Dn, the dihedral group of order 2n, further in §8. ◄ 3 Note 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 an element of S4 by considering the final position of each vertex after applying the symmetry. For example, (counterclockwise) rotation by 90° will take vertex 1 to position 2, vertex 2 to position 3, etc. Hence we may represent a by the permutation (23³1) The reader may check that under this correspondence the product of two symmetries will be represented by the product of the two corresponding permutations. It is often useful to represent the elements of a group by permutations on some set and, as we will see, this is always possible.
Transcribed Image Text:58 111/Semigroups, Monoids, and Groups ► (1.6) EXAMPLE: D₁, the Symmetries of a Square. Although the study of groups usually seems quite abstract at first, much of the motivation for groups comes from geometric problems, especially prob lems 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 that the square covers the same region at the end of the motion as it did at the start. More precisely, if we take axes for 3-space with the origin at the center 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 a rotation, or a reflection about some line, or a composition of rotations and reflections. We will assume all rotations below to be in the counterclockwise direction. (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 regular plane n-gon (i.e., a polygon in the plane with n equal sides and each exterior angle equal to 360°/n). Figure 1.1 For 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 given by reflection about the line h. Again, composition of mappings is an obvious operation 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 mapping a = rotation by 90° a² rotation by 180° a³ rotation by 270° 3= reflection about the line h aß= reflection about the diagonal BD a²3 reflection about the line v a³3= reflection about the diagonal AC. 1/Examples It is not too hard to determine the following multiplication table for D4- (1.8) TABLE ( a 0² 03 B aß 0²6 a³6 € 9² a³ B aå a²8 a³8 a 0² 9³ € 0³3 B aß a²3 0² a² Qª C a a²3 a³3 B Q³ a³ C 0² aß a²8 a³3 B aß 28²8265 aß a²8 aß a²3 a³8 a³8 3 aß a²B B 9³ € a 0² 93 9² qª C a Figure 1.2 a³B aB B 59 a 9² 03 One 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 group of order 8. We note that each reflection is its own inverse, as is rotation by 180°; rotation by 90° and rotation by 270° are inverse to each other. We will study Dn, the dihedral group of order 2n, further in §8. ◄ 3 Note 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 an element of S4 by considering the final position of each vertex after applying the symmetry. For example, (counterclockwise) rotation by 90° will take vertex 1 to position 2, vertex 2 to position 3, etc. Hence we may represent a by the permutation (23³1) The reader may check that under this correspondence the product of two symmetries will be represented by the product of the two corresponding permutations. It is often useful to represent the elements of a group by permutations on some set and, as we will see, this is always possible.
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