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(a) Describe geometrically the symmetries of P represented in cycle form by (24)(57) and (1 8) (2 7)(3 6)(4 5). (b) Write down all the symmetries of P in cycle form as permutations of {1, 2, 3, 4, 5, 6, 7, 8)}, and describe geometrically each symmetry that is a rotation or reflection. (c) Write down all the conjugacy classes of the symmetry group S(P) of P.

(a) Describe geometrically the symmetries of P represented in cycle form by (24)(57) and (1 8) (2 7)(3 6)(4 5). (b) Write down all the symmetries of P in cycle form as permutations of {1, 2, 3, 4, 5, 6, 7, 8)}, and describe geometrically each symmetry that is a rotation or reflection. (c) Write down all the conjugacy classes of the symmetry group S(P) of P.

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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The solid P shown below is composed of an equilateral triangular prism and two tetrahedra joined together. All the edges have the same length, and one face of each tetrahedron coincides with a triangular face of the prism. The locations of the vertices are numbered so that the symmetries of P can be represented as permutations of the set {1, 2, 3, 4, 5, 6, 7, 8). 5 - 4 7 6 P (a) Describe geometrically the symmetries of P represented in cycle form by (2 4) (57) and (1 8) (2 7)(3 6) (45). (b) Write down all the symmetries of P in cycle form as permutations of {1, 2, 3, 4, 5, 6, 7, 8}, and describe geometrically each symmetry that is a rotation or reflection. (c) Write down all the conjugacy classes of the symmetry group S(P) of P. (d) Determine a normal subgroup of S(P) of order 2, a normal subgroup of order 3, and a normal subgroup of order 6, justifying your answers.
Transcribed Image Text:The solid P shown below is composed of an equilateral triangular prism and two tetrahedra joined together. All the edges have the same length, and one face of each tetrahedron coincides with a triangular face of the prism. The locations of the vertices are numbered so that the symmetries of P can be represented as permutations of the set {1, 2, 3, 4, 5, 6, 7, 8). 5 - 4 7 6 P (a) Describe geometrically the symmetries of P represented in cycle form by (2 4) (57) and (1 8) (2 7)(3 6) (45). (b) Write down all the symmetries of P in cycle form as permutations of {1, 2, 3, 4, 5, 6, 7, 8}, and describe geometrically each symmetry that is a rotation or reflection. (c) Write down all the conjugacy classes of the symmetry group S(P) of P. (d) Determine a normal subgroup of S(P) of order 2, a normal subgroup of order 3, and a normal subgroup of order 6, justifying your answers.
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