总述
Lectures of Visual Group Theory
Visual Group Theory
Overview
Three nonstandard aspect of nonstandard approach in the book:
- images and visual examples-the heart
- focuses more on finite groups than infinite ones-easier to diagram and give a solid foundation of intuition
- approaches groups from the opposite direction of traditional textbooks-groups are defined to be collections of actions and later proved to be sets with binary operations
- linear order, to be read in order
- exceptions: Chapter 5 and Chapter 10, you can skip most of them without understanding other chapters. But the definition of abelian in Section 5.2 and the Cauchy's Theorem in Section 9.2 are important
What is a group?
Group theory is not primarily about numbers, but rather about patterns and symmetry
Four key observations
Observation 1
- There is a predefined list of moves that never changes.
Observation 2
- Every move is reversible.
Observation 3
- Every move is deterministic.
Observation 4
- Moves can be combined in any sequence.
A move is a twist of one of the six faces.
Let's rephrase the 4 observations as rules (axioms) that will define the boundaries of our objects of study.
Rules of groups
Rule 1
- There is a predefined of actions that never changes.
Rule 2
- Every action is reversible.
Rule 3
- Every action is deterministic.
Rule 4
- Any sequence of consecutive actions is also an action.
We swapped the word "move" for "action"
The (usually short) list of actions required by Rule 1 is our set of building blocks; called the generators.
Not all actions are generators. Generators are the subset of all possible actions in a group.
Rule 4 tells us that any sequence of the generators is also an action.
Our unofficial definition of a group (We'll make things a bit rigorous later.)
Definition (informal)
A group is a set of actions satisfying Rules 1-4.
Let's call these generators a, b, c, d, e, and f. Every word over the alphabet \(\{a, b, c, d, e, f\}\) describes a unique configuration of the cube (starting from the solved position)
Obviously, I don't care which is which.
Summary of the big ideas
Cayley graphs
A road map for the Rubik's Cube
Let's pretend for a moment that we were interested in writing a complete solutions manual for the Rubik's Cube. And let me be more specific
We can think of the Big Book as a road map for the Rubik's Cube. Each page says "you are here" and “if you follow this road, you'll end up over there."
Despite the Big Book's apparent shortcomings, it made for a good thought experiment. The Big Book is a map of a group. The Big Book introduced the mapmaking ideas, although the map is too large. We can use the same ideas to map out any group. In fact, we shall frequently do exactly that.
And Let's try something simpler....
The Rectangle Puzzle
How can we check the moves of the Rectangle Puzzle form a group?
More on arrows
node \(y\) is the result of applying the action $ g G $ to node $x $.
if an action \(h \in G\) is its own inverse (that is, \(h ^ 2 = e\)), then we have a 2-way arrow. For clarity, our convention is to drop the tips on all 2-way arrows like that
When we focus on a group's structure, we frequently omit the labels at the nodes
isomorphic: corresponding or similar in form and relations.
The Klein 4-group
Any group with the same Cayley diagram as the Rectangle Puzzle and the 2-Light Switch Group is called the Klein 4-group, denoted by \(V_4\) for vierergruppe, “four-group" in German. It is named after the mathematician Felix Klein.
The triangle puzzle
Properties of Cayley graphs
A Theorem and Proof!
Theorem Suppose an action \(g\) has the property that \(gx = x\) for some other action \(x\). Then g is the identity action, i.e., \(gh = h = hg\) for all other actions \(h\).
\(g\) apply to anything else !
Proof:
The identity action (we'll denote by 1) is simply the action \(hh^{-1}\), for any action \(h\). If \(gx = x\), then multiplying by \(x^{-1}\) on the right yields:
\[g = gxx^{-1} = xx^{-1} = 1\].
Thus \(g\) is the identity action.
Groups in science, art, and mathematics
Group of symmetries
How to make a group out of symmetries
Groups relate to symmetry because an object's symmetries can be described using arrangements of the object's parts.
Algorithm 1. Identify all the parts of the object that are similar (e.g., the corners of an n-gon), and give each such part a different number. 2. Consider the actions that may rearrange the numbered parts, but leave the object it the same physical space. (This collection of actions forms a group. 3. (Optional) If you want to visualize the group, explore and map it as we did in the previous lecture with the rectangle puzzle, etc.
Footprint: the physical space that an object occupies
This section seems to be useless. So if I have time later, let's continue to see at 9:37.
Group presentations
Our definition of groups before (a collection of actions that obeyed Rules 1-4) is not the ordinary definition of a group. We'll be working toward introducing the standard (and more formal) definition of a group. Along the way, we will learn some helpful tools to get us there.
Group presentations, an algebraic device to concisely describe groups by their generators and relations.
More on Cayley diagrams
All arrows of a fixed color correspond to the same generator.
Two things with the node:
Labeling the nodes with configurations of a thing we are acting on
Leaving the nodes unlabeled (this is the "abstract Cayley diagram”)
Label the nodes with actions (this is called a “diagram of actions").
motivated by the fact that every path in the Cayley diagram represents an action of the group
Motivating idea If we distinguish one node as the “unscrambled” configuration and label that with the identity action, then we can label each remaining node with the action that it takes to reach it from the unscrambled state.
A "group calculator"
A concise way to describe \(V_4\) is the following group representation: \[ V_4 = <v, h | v^2 = e, h^2 = e, vh = hv > \] The following is one (of many!) presentations for this group: \[ D_3 = <r, f | r^3 = e, f^2 = e, r^2f = fr> \]
To be more precise
If we want to be more precise, we use a group
presentation of the following form: \[
G = <generators | relations>
\] The vertical bar here can be thought of as meaning "subject
to".
The famous "halting problem" is unsolvable
Multiplication tables
We are almost ready to introduce the formal definition of a group In this lecture, we will introduce one more useful algebraic tool for better understanding groups: multiplication tables. We will also look more closely at inverses of the actions in a group. Finally, we will introduce a new group of size 8 called the quaternions which frequently arise in theoretical physics.
Inverses
denoted by \(g^{-1}\)
Multiplication tables
follow the convention that write row \(g\) before column \(h\)
Some remarks
A group is abelian \(iff\) its multiplication table is symmetric about the “main diagonal."
In each row and each column, each group action occurs exactly once.
The following theorem will explain that
Theorem: An element cannot appear twice in the same row or column of a multiplication table.
Proof:
Suppose that in row \(a\), the element \(g\) appears in columns \(b\) and \(c\). Algebraically, this means \[ ab=g=ac \] Multiplying everything on the left by \(a^{-1}\) yields \[ a^{-l}ab=a^{-1}g=a^{-1}ac \Rightarrow b = c \] Thus, \(g\) (or any element) element cannot appear twice in the same row.
Similarly, we can finish the proof for column.
The quaternion group
The formal definition of a group
\(S\) is closed on the operation *
Associativity
交换律
An operation is associative if parentheses are permitted anywhere, but required nowhere.
Formal definitions
Viewing groups from these two different paradigms:
- a group as a collection of actions
- a group as a set with a binary operation
Properties
Theorem: Every element of a group has a unique inverse
Proof:
Theorem: Every group has a unique identity element
Cyclic and abelian groups
Overview
\(5\) families of groups:
- cyclic groups
- abelian groups
- dihedral groups
- symmetric groups
- alternating groups
Informally, a group is cyclic if it is generated by a single element, and is abelian if multiplication commutes. (Like \(a * b = b * a, \forall a, b \in G\))
Cyclic groups, additively
Definition: A group is cyclic if it can be generated by a single element.
Finite cyclic groups describe the symmetry of objects that have only rotational symmetry.
\[ C_n = <r| r^n = e> \]
Remark: This is a natural generator, but not the only one. For instance, \(r^3\) is also a generator.
Definition: The order of a group \(G\) is the number of distinct elements in \(G\), denoted by \(|G|\).
The cyclic group of order \(n\) (i.e., \(n\) rotations) is denoted \(C_n\) (or sometimes by \(\mathbb{Z}_n\)). In fact, the alternative notation \(\mathbb{Z}_n\) comes from the fact that the binary operation in \(C_n\) is just modular addition. To add two numbers in \(\mathbb{Z}_n\), add them as integers, divide by \(n\), and take the remainder.
Cyclic groups, multiplicatively
If \(r\) is a generator (e.g., a rotation by \(2 \pi /n\)), then we can denote the \(n\) elements by \[ 1, r, r^2, \dots , r^{n-1} \] Think of \(r\) as the complex number \(e^{2\pi i/n}\), with the group operation being multiplication!
复数相乘，模相乘，辐角相加
More on cyclic groups
The (unique) infinite cyclic group (additively) is (\(\mathbb{Z}, +\)), the integers under addition. Using multiplicative notation, the infinite cyclic group is \[ G=<r\quad|\quad>=\{r^k : k \in \mathbb{Z}\} \] For the infinite group \((\mathbb{Z}, +)\), only \(1\) or \(-1\) can be generators. (Considering multiple generators are pointless.)
Proposition: Any number from \(\{0, 1, \dots, n-1\}\) that is relatively prime to \(n\) will generate \(\mathbb{Z}_n\).
For example, \(1, 2, 3, 4\) all generate \(\mathbb{Z}_5\). i.e., \[ \mathbb{Z}_5 = <1> = <2> = <3> = <4>. \]
The above notation just means "generated by" instead of a presentation.
Observation: One of the most important properties of the multiplication tables is that, if the headings on the multiplication table are arranged in the "natural" order \((0, 1, 2, \dots , n-1)\) or \((e, r, r^2, \dots, r^{n-1})\), then each row is a cyclic shift to the left of the row above it.
Orbits
Orbits are usually written with braces. In the case, the orbit of \(r\) is \(\{e, r, r^2\}\), and the orbit of \(f\) is \(\{e, f\}\)
Definition: The order of an element
Anything you can get by the path, following or backwards
