MATH 111 m2

MATH 111 m2 - MODEL ANSWERS TO THE SECOND HOMEWORK 1. Label...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MODEL ANSWERS TO THE SECOND HOMEWORK 1. Label the vertices of the square A , B , C , D , where we start at the top left hand corner and we go around the square clockwise. In particular A is opposite to C and B is opposite to D . There are three obvious types of symmetries. There are rotations. One obvious rotation R corresponds to rotation clockwise through / 2 radians. The others are R 2 , R 3 and the identity I . They represent rotation through and 3 / 2. There are two sorts of flips. One set of flips are diagonal flips. The first D 1 fixes the diagonal AC and switches B and D . The other D 2 fixes the diagonal BD and A and C . The other possibility is to look at the flip F 1 which switches A and D and B and C and the flip F 2 which switches A and B and C and D . I claim that this exhausts all possible symmetries. In fact any symme- try is determined by its action on the fours vertices A , B , C and D . Now there are 24 = 4! possible such permutations. On the other hand any symmetry of a square must fix opposite cor- ners. Thus once we have decided where to send A , for which there are four possibilities, the position of C is determined, it is opposite to A . There are then two possible positions for B . So there are at most eight symmetries and we have listed all of them. We start looking for subgroups. Two trivial examples are D 4 and { I } . A non-trivial example is afforded by the set of all rotations { I, R, R 2 , R 3 } . Clearly closed under products and inverses. Note that rotation through radians R 2 generates the subgroup { I, R 2 } . Simliarly, since any flip is its own inverse, the following are all sub- groups, { I, F 1 } , { I, F 2 } , { I, D 1 } and { I, D 2 } . Now try combining side flips and diagonal flips. Now F 1 D 1 = R 3 . So any subgroup that con- tains F 1 and D 1 must contain R 3 and hence all rotations. From there it is easy to see we will get the whole of G . So we cannot combine side flips with diagonal flips. Now consider combining rotations and flips. Note that F 1 F 2 = R 2 and D 1 D 2 = R 2 by direct computation. We then try to see if { I, F 1 , F 2 , R 2 } is a subgroup. As this is finite, it suffices to check that it is closed underis a subgroup....
View Full Document

Page1 / 5

MATH 111 m2 - MODEL ANSWERS TO THE SECOND HOMEWORK 1. Label...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online