Each of figures A and B below is obtained by adding a straight line to a regular hexagon. In figure A the straight line joins two opposite vertices; in figure B the straight line joins the midpoints of two opposite edges. 2 1 6 5 2 1 6 5 3 4 A 3 4 B Let S(A) and S(B) be the symmetry groups of figures A and B, respectively. Two elements of S(A) are a = (14) (25) (36) and r = (1 4)(2 3) (5 6). (a) Are a and r conjugate in the group S(A)? Justify your answer. (b) Are a and r conjugate in the group S6? Justify your answer. (c) List all the elements of S(A), and list all the elements of S(B), representing all the symmetries as permutations, in cycle form, of the six vertex labels. (d) Are S(A) and S(B) conjugate in the group S(O)? Justify your answer. (e) Are S(A) and S(B) isomorphic? Justify your answer.
Each of figures A and B below is obtained by adding a straight line to a regular hexagon. In figure A the straight line joins two opposite vertices; in figure B the straight line joins the midpoints of two opposite edges. 2 1 6 5 2 1 6 5 3 4 A 3 4 B Let S(A) and S(B) be the symmetry groups of figures A and B, respectively. Two elements of S(A) are a = (14) (25) (36) and r = (1 4)(2 3) (5 6). (a) Are a and r conjugate in the group S(A)? Justify your answer. (b) Are a and r conjugate in the group S6? Justify your answer. (c) List all the elements of S(A), and list all the elements of S(B), representing all the symmetries as permutations, in cycle form, of the six vertex labels. (d) Are S(A) and S(B) conjugate in the group S(O)? Justify your answer. (e) Are S(A) and S(B) isomorphic? Justify your answer.
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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Transcribed Image Text:Each of figures A and B below is obtained by adding a straight line to a
regular hexagon. In figure A the straight line joins two opposite vertices; in
figure B the straight line joins the midpoints of two opposite edges.
2
1
6
5
2
1
6
5
3
4 A
3 4 B
Let S(A) and S(B) be the symmetry groups of figures A and B, respectively.
Two elements of S(A) are a = (14) (25) (36) and r = (1 4)(2 3) (5 6).
(a) Are a and r conjugate in the group S(A)? Justify your answer.
(b) Are a and r conjugate in the group S6? Justify your answer.
(c) List all the elements of S(A), and list all the elements of S(B),
representing all the symmetries as permutations, in cycle form, of the
six vertex labels.
(d) Are S(A) and S(B) conjugate in the group S(O)? Justify your answer.
(e) Are S(A) and S(B) isomorphic? Justify your answer.
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