MIT 18.703 Modern Algebra / Assignment 3

[E7-87-83]

Course: MIT 18.703 Modern Algebra
Document: ./MIT/Solutions/18.703/Assignment3/
Date: Oct 2023

Ex9 對你來講太trivial XD無做呢？

Ex8

＃44
consider a regular plane n-gon for n ≥ 3. Each way that two copies of such an n-gon can be placed, with one covering the other, corresponds to a certain permutation of the vertices. The set of these permutations is a group, the nth dihedral group Dn , under permutation multiplication. Find the order of this group Dn . Argue geometrically that this group has a subgroup having just half as many elements as the whole group has.

唔知本書前面有無講一定係rotation或reflection。（唔知有無講 plane isometry 不外乎係 translations, reflections 或 rotations 。）因為佢提到 permutation of the vertices, 我諗到可以咁樣prove:
Given two distinct vertices A、 X, 最多只有一個vertex Y distinct from X of which distance of A and X equals distance of A and Y 。所以，for any permutation p of the n-gon mapping A to A and satisfying our requirement, either p(X) = X or p(X) = Y.

Choose X as one of the clockwise adjacent vertices of A. Y must be the other adjacent vertex of A. For any permutation p of the n-gon mapping A to A and satisfying our requirement, for the case p(X) = X, [省略 induction 過程, 主要係argue distance，嚴謹證明需要] by induction along the clockwise direction of vertices starting from A, p is the identity permutation; for the case p(X) = Y, [省略 induction 過程, 主要係argue distance，嚴謹證明需要] by induction along the clockwise direction of vertices starting from A, vertices X, X', X'', X'''... , p(X') = Y', p(X'') = Y'', p(X''') = Y'''... and X',Y' distinct, X'',Y'' distinct... and Y', Y'', Y'''... are pairwise adjacent.

Choosing a vertex A* distinct from A. For every vertex B distinct from A, 最多只有一個vertex M distinct from B of which distance of A and B equals distance of A and M 。For any permutation p of the n-gon mapping A to A* and satisfying our requirement, d(A, B) = d(A*, p(B)) and d(A, M) = d(A*, p(M)) , 又，因為 d(A, B) = d(A, M) ， d(A*, p(B)) = d(A*, p(M)) ， either p(B) = p(M), or p(B) is the reflected point of p(M) of reflection along an axis passing through A. 後者係一個reflection，前者[省略induction過程, 又係argue along a clockwise direction of vertices starting from A]係一個rotation 。 [未完]
【寫到自己都煩...】

【重要觀察】因為 covering 都 map centre of regular n-gon back to the centre of regular n-gon，【其實應該仲要補充........】所有呢啲 permutations 全部都係 rotations 同埋 reflections along axes passing through the centre.

我嘅做法或者你這裏的做法，都需要補充證明 rotation * reflection along a chosen axis = reflection along another axis 以及 reflection along a chosen axis * rotation = reflection along yet another axis (* for composition)

至於 associativity, 則可以說 follows from permutation group 。

要話 the set of rotations 係subgoup, 需要補充一句 rotation * rotation' = rotation'' 同埋 rotation * (rotation' * rotation'') = (rotation * rotation') * rotation''。但本書前面好似已經講到cyclic groups。

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【REF: 《Algebra》 by Artin, section 6.4】
Theorem 6.4.1 Let G be a finite subgroup of the orthogonal group O2. There is an integer n such that G is one of the following groups:

(a) Cn: cyclic group of order n generated by the rotation angle 2\pi / n.

(b) Dn: dihedral group, generated by rotation angle 2\pi / n , and a reflection r' about a line l through the origin.

Proof: 2x2 matrix operation.

然後書上嘅寫法：A regular n-gon is carried to itself by the rotation by 2 \pi / n about its center, and also by some reflections. Theorem 6.4.1 identifies the group of all symmetries as Dn. 【我自己仲覺得嚴謹性差少少。】

【唉……唔滿意 .＿. 】

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＃45

Consider a cube that exactly fills a certain cubical box. As in Examples 8.7 and 8.10, the ways in which the cube can be placed into the box correspond to a certain group of permutations of the vertices of the cube. This group is the group of rigid motions (or rotations) of the cube. How many elements does this group have? Argue geometrically that this group has at least three different subgroups of order 4 and at least four different subgroups of order 3.

three different subgroups of order 4 - each is a cyclic group mapping a square face back to itself and the respective opposite square face to that opposite square face

four different subgroups of order 3 - Yes, rotation along the (body) diagonal; each is a cyclic group mapping the equilateral triangle back to itself

ref image: https://www.geogebra.org/resource/jkM4DVgs/TKXTm2hjVoDqT4Qt/[email protected]

Bonus Problems
可以讀 Artin Section 6.12