3. As I mentioned one night in class, symmetries of objects often give groups. Here you will construct such a group. Draw a square and label its vertices 1, 2, 3, 4 going clockwise. It turns out that the square has eight symmetries. There are four rotations (you need to tell me the angles) and four reflections (you need to draw the line of symmetry of each). Each of these eight symmetries can be thought of as an element of S4 (according to how it permutes the numbered vertices). For each of the eight symmetries, tell me the element of S4 that you get! (Note: this produces an eight-element subgroup of S4 that you might have had trouble guessing otherwise.)
3. As I mentioned one night in class, symmetries of objects often give groups. Here you will construct such a group. Draw a square and label its vertices 1, 2, 3, 4 going clockwise. It turns out that the square has eight symmetries. There are four rotations (you need to tell me the angles) and four reflections (you need to draw the line of symmetry of each). Each of these eight symmetries can be thought of as an element of S4 (according to how it permutes the numbered vertices). For each of the eight symmetries, tell me the element of S4 that you get! (Note: this produces an eight-element subgroup of S4 that you might have had trouble guessing otherwise.)
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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Question
Transcribed Image Text:3. As I mentioned one night in class, symmetries of objects often give groups. Here you
will construct such a group.
Draw a square and label its vertices 1, 2, 3, 4 going clockwise.
It turns out that the square has eight symmetries. There are four rotations (you need to
tell me the angles) and four reflections (you need to draw the line of symmetry of each).
Each of these eight symmetries can be thought of as an element of S4 (according to how
it permutes the numbered vertices). For each of the eight symmetries, tell me the element of
S4 that you get!
(Note: this produces an eight-element subgroup of S4 that you might have had trouble
guessing otherwise.)
Expert Solution
Step 1
We will find 4 symmetries of square using rotations and 4 symmetries using reflection and then will identify the permutations of S_4 that occur.
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