Question

What am I supposed to do?
Let S be a square with vertices 1,2,3 and 4 initially placed, respectively, at the points (-1,1),(1,1),(1,-1) and )-1,-1) in the xy plane. Let r be the symmetry of S produced by a counterclockwise rotation of pi/2 radians, and let s be the symmetry of S corresponding to reflection about the x-axis. Show that all eight symmetries of S can be written in the form \(r^is^j\), where i\(\in\){0.1.2.3}, and j\(\in\) {0,1}
Here \(r^0 =e\), \(r^1 =r\), \(r^2 = rr\), \(r^3= rrr\), \(s^0=e\) and \(s^1 =s\).
Please, help

Answer 1

@Kainui

Answer 2

@precal

Answer 3

sorry this is out of my realm

Answer 4

Thanks for reply though.:)

Answer 5

@dumbcow

Answer 6

what level of math is this? Just curious

Answer 7

abstract algebra

Answer 8

sorry i am unfamiliar with the notation for symmetries.

Answer 9

@BSwan @ikram (in case you came online under those name, hihihi) you see!! they all ignore me. ha!!

Answer 10

@kirbykirby

Answer 11

*

Answer 12

I'm not 100% sure of this proof... I haven't done much in group theory stuff.

Imagine you have a square and you label the vertices with ABCD. I believe what you have is the dihedral group \(D_4\) in group theory. Now, imagine what the 8 symmetries are... there will be 4 rotations
A B --> identity (1)
C D

B D --> rotate 90 (2)
A C

D C --> rotate 180 (3)
B A

C A --> rotate 270 (4)
D B

And 4 axes of symmetry:
C D --> flip along x-axis (5)
A B

B A --> flip along y-axis (6)
D C

A C --> flip along diagonal \ (7)
D B

D B --> flip along diagonal / (8)
C A

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Note that (1), (2), (3), and (4) are \(r^0, r^1, r^2, r^3\) respectively. Now try composing each other those with each symmetry, \(s^0\) and \(s^1\). Now \(s^0\) is the identity, so all the notations followed by \(s^0\) will give the same figure itself. Now, see that you obtain (5), (6), (7), (8) by doing the x-axis symmetry \(s^1\) with each of (1), .., (4).

Answer 13

@kirbykirby Thanks a ton. I know what I am supposed to do now. Again, thank you very much. I am new in the field so that so many "language" I don't understand. :)

Answer 14

Yw :) Yes abstract algebra takes a bit of time to get used to :)