The figure F, shown below, is a square-based pyramid with edges all of equal length. The location of the top vertex is labelled 1, and the locations of the four vertices of the base are labelled 2, 3, 4 and 5. 3 1 2 F 5 (a) Write down, in cycle form, the elements of the symmetry group S(F). Denote each symmetry by a single letter and describe it geometrically. (b) Write down a group table for S(F), using your letters from part (a). (c) State to which one of the following groups S(F) is isomorphic, and show that S(F) is not isomorphic to either of the other two groups: S4, (Z8, +8), S(O). (d) Write down one subgroup of S(F) of order 2, and one subgroup of order 4.
The figure F, shown below, is a square-based pyramid with edges all of equal length. The location of the top vertex is labelled 1, and the locations of the four vertices of the base are labelled 2, 3, 4 and 5. 3 1 2 F 5 (a) Write down, in cycle form, the elements of the symmetry group S(F). Denote each symmetry by a single letter and describe it geometrically. (b) Write down a group table for S(F), using your letters from part (a). (c) State to which one of the following groups S(F) is isomorphic, and show that S(F) is not isomorphic to either of the other two groups: S4, (Z8, +8), S(O). (d) Write down one subgroup of S(F) of order 2, and one subgroup of order 4.
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:The figure F, shown below, is a square-based pyramid with edges all of equal
length. The location of the top vertex is labelled 1, and the locations of the
four vertices of the base are labelled 2, 3, 4 and 5.
1
3
2
F
5
(a) Write down, in cycle form, the elements of the symmetry group S(F).
Denote each symmetry by a single letter and describe it geometrically.
(b) Write down a group table for S(F), using your letters from part (a).
(c) State to which one of the following groups S(F) is isomorphic, and
show that S(F) is not isomorphic to either of the other two groups:
S4, (Z8, +8), S(O).
(d) Write down one subgroup of S(F) of order 2, and one subgroup
of order 4.
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