Algebraic Structures

(代数系)

Discrete Mathematics I

11th lecture, December 16, 2018

https://www.sw.it.aoyama.ac.jp/2019/Math1/lecture11.html

Martin J. Dürst

© 2006-19 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

• Remaining schedule, final exam
• Summary, leftovers, and homework from last lecture
• Algebraic Structures
• Groups
  • Group axioms
  • Examples of groups
  • Permutations and symmetric groups
  • Simple group theorems
  • Group isomorphisms
  • Cayley tables

Remaining Schedule

• December 16: this lecture
• December 20: 12th lecture
• January 10: 13th lecture
• January 17: 14th lecture (makeup class)
• January 24: 15th lecture
• Janualy 31, 11:10-12:35: Term final exam

About makeup classes: The material in the makeup class is part of the final exam. If you have another makeup class at the same time, please inform the teacher as soon as possible.

補講について: 補講の内容は期末試験の対象。補講が別の補講とぶつかる場合には事前に申し出ること。

Final Exam・期末試験

Coverage:

Complete contents of lecture and handouts

Past exams: 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018

How to view example solutions:

Press the [表示 (S)] button or the [S] key. To revert, press the [非表示 (P)] button or press the [P] key.
Sometimes, more than one key press is needed to start switching.
Some images and example solutions are missing.

Important Points about Final Exam

• Most problems are in Japanese and English
• Most answers can be in Japanese or English
• Read problems carefully (distinguish between calculation, proof, explanation,...)
• Be able to explain concepts in your own words
• Be able to do calculations (base conversions, truth tables,...) speedily
• Combine and apply knowledge from different lectures
• Write clearly

Leftovers of Last Lecture

Hasse diagrams, equivalence relations and order relations in matrix representation

Summary of Last Lecture

We defined the following properties of binary relations:

1. Reflexive: ∀x∈A:xRx; ∀x∈A: (x, x) ∈ R
2. Symmetric: ∀x, y ∈A: xRy ⇔ yRx;
   ∀x, y ∈A: (x, y) ∈ R ⇔ (y, x) ∈ R
3. Antisymmetric: ∀x, y ∈A: xRy ∧ yRx ⇒ x=y
4. Transitive: ∀x, y, z ∈A: xRy ∧ yRz ⇒ xRz

A relation that is reflexive, antisymmetric, and transitive is a (partial) order relation.

A relation that is reflexive, symmetric, and transitive is an equivalence relation.

(Partial) order relations can be represented with Hasse diagrams.

Last Week's Homework:
Combinations of Properties of Relations

Investigate all combinations of the four properties of relations introduced in this lecture (reflexive, symmetric, antisymmetric, transitive). For each combination, give an example relation on the minimum size set possible, or explain why such a combination is impossible. Present the 16 combinations in a table similar to the tables used in the homework of lecture 4. In the table, also include a column with the minimum size of the set on which the example relation is formed. Use {b}, {b, c}, and {b, c, d} for sets with one, two, and three elements, respectively.

Hint: Two combinations are impossible. One combination is possible with a relation on an empty set. One combination is possible with a relation on a set of size one. Four combinations are possible with a relation on a set of size two. The other combinations need a relation on a set of size three.

Homework Solution

Explanation why two combinations are impossible:
In both cases, the relations need to be symmetric and antisymmetric, but not transitive. In relations that are both symmetric and antisymmetric, all positions except those on the (main) diagonal are false. Such relations are automatically transitive (because they are their own transitive closure).

Algebraic Structure

Very general view on mathematical objects

An algebraic structure is a class of mathematical objects that all share the same general structure.

Properties shared by all algebraic structures are:

• A set (or more than one set)
• An operation on the elements of the set
  (more than one operation in some cases)
  Condition: The results of the operation(s) also have to be elements of the set
  This is called closure; the set is closed under the operation
  
• Some axioms
• Proofs of theorems and properties from the axioms

Previously Encountered Examples

• Plane geometry and the Euclidean axioms
• Natural numbers and the Peano axioms
• Truth values (true/false) and the axioms of basic logic (several choices)

• Up to now:
  
  • Start with actual objects
  • Try to axiomatize
• Today:
  
  • Try to find a small set of "interesting", "productive" axioms
  • Look at commonalities among different sets and different operations

Example of Algebraic Structure: Group

• One set (A)
• One binary operation (•; the set is closed under the operation: ∀b,c∈A: b•c∈A)
• Three axioms:
  1. Associativity (∀b,c,d∈A: (b•c)•d = b•(c•d))
  2. (Existence of a) identity element e (∃e∈A: ∀b∈A: e•b = b = b•e)
  3. (Existence of an) inverse element b' (∀b∈A: ∃b'∈A: b•b' = e = b'•b)
     (The inverse element may also be written b^-1)
• Note: Commutativity is not necessary

The Integers with Addition as a Group (ℤ, +)

• Set: ℤ (integers)
• Operation: + (addition)
• Associativity: ∀b,c,d∈ℤ: (b+c)+d = b+(c+d)
• Identity element: 0
• Inverse element: b' = -b

The Reals with Multiplication as a Group (ℝ-{0}, ·)

• Set: ℝ-{0} (real numbers without 0)
• Operation: · (multiplication)
• Associativity: ∀b,c,d∈(ℝ-{0}): (b·c)·d = b·(c·d)
• Identity element: 1
• Inverse element: b' = 1/b (inverse/reciprocal, b^-1)

The Positive Reals with Multiplication as a Group (ℝ⁺, ·)

• Set: ℝ⁺ (positive real numbers)
• Operation: · (multiplication)
• Associativity: ∀b,c,d∈ℝ⁺: (b·c)·d = b·(c·d)
• Unit element: 1
• Inverse element: b' = 1/b

Permutations

• There are n! permutations of elements from a set S with size |S|=n
• Permutations can be seen as ordered selections
  Example: From the set {Aoyama, Sagamihara} we can create the permutations (Aoyama, Sagamihara) and (Sagamihara, Aoyama)
  Example: From the set {cat, dog, horse, cow}, we can select the permutation (dog, cow, cat, horse) (and 23 others)

Permutations as Exchanges

• Permutations can be seen as ways to exchange elements
  Example: For a tuple/list with two elements, there are two permutations:
  
  1. One permutation that keeps the same order: (1, 2)
  2. One permutation that changes the order of the elements: (2, 1)
• We denote such permutations by assuming we start with a tuple of the first n integers ((1, 2,...)), and show the result of the permutation
• Example: The tuple (cat, dog, horse, cow), when permuted with the permutation (2, 4, 1, 3), results in (dog, cow, cat, horse)

Composition of Permutations

• When seen as exchanging elements, permutations can be composed
• We use ∘ to denote composition
• Example: (2, 4, 1, 3) ∘ (1, 4, 2, 3) = (2, 3, 4, 1)
• Composition of permutations can be show by using cards
• Cut out and use the cards at permutations.svg

Permutations as a Group

• The permutations of sets of size n together with composition form a group:
  • All compositions of permutations result in another permutation
  • Permutations are associative
  • The identity element is (1, 2, 3, 4, ...)
  • Each permutation has an inverse
    Example: The inverse of (2, 4, 1, 3) is (3, 1, 4, 2)
  • Commutativity does not hold
    Example: (2, 4, 1, 3) ∘ (1, 4, 2, 3) = (2, 3, 4, 1)
    (1, 4, 2, 3) ∘ (2, 4, 1, 3) = (4, 3, 1, 2)
• These groups are called symmetric groups of order n

Group Theorem: Uniqueness of Identity

Existence of identity element (axiom): ∃e∈A: ∀b∈A: e•b = b = b•e

Theorem: The identity element of a group is unique
(∃c∈A: ∀x∈A: c•x = x) ⇒ c = e

Another way to express this: There is only one identity element
(|{c|c∈A, ∀x∈A: c•x = x)}|= 1)

Proof:

c•x = x [inverse axiom, closure]

(c•x)•x' = x•x' [associativity axiom]

c•(x•x') = x•x' [inverse axiom, on both sides]

c•e = e [identity axiom]

c = e Q.E.D. (similar proof for right idenity)

Group Theorem: Uniqueness of Inverse

Existence of an inverse (axiom): ∀b∈A: ∃b'∈A: b•b' = e = b'•b

Theorem: Each inverse is unique
∀a, b∈A: (a•b = e ⇒ b=a')

Proof:

a•b = e [applying a'• on the left]

a'•(a•b) = a'•e [associativity axiom]

(a'•a)•b = a'•e [inverse axiom]

e•b = a'•e [identity axiom, on both sides]

b = a' Q.E.D. (similar proof for left inverse)

Group Theorem: Cancellation Law

Theorem: ∀a, b, c ∈A: (a•c = b•c ⇒ a=b)

Proof:

a•c = b•c [applying c' on the right]

(a•c)•c' = (b•c)•c' [associativity]

a•(c•c') = b•(c•c') [inverse axiom, on both sides]

a•e = b•e [identity axiom, on both sides]

a = b Q.E.D. (similar proof for left cancellation)

Group Isomorphism

• Isomorphism is a one-to-one correspondence
• Two groups (G, •) and (H, ∘) are isomorphic if there is a function f so that:
  • ∀h∈H: ∃g∈G: h = f(g)
  • ∀g₁, g₂∈G:g₁≠g₂ → f(g₁)≠f(g₂)
  • ∀g₁, g₂∈G: f(g₁•g₂) = f(g₁)∘f(g₂)

  The elements in G and H correspond one-to-one, and the operation works exactly the same on corresponding elements.

• If two groups are isomorphic
  
  • They have the same number of elements (|G|=|H|)
  • They have the same structure
  • From a mathematical viewpoint, they can be considered to be the same

Examples of Isomorphic Groups

• Example 1: (ℝ, +) is isomorphic to (ℝ⁺, ·), with f(x) = a^x (a>1)
• Example 2: Three isomorphic groups (shown as Cayley tables)

Cayley Tables

• Finite groups are usually described using Cayley tables
• Cayley tables look very much like multiplication tables
• Conventions:
  • The left operands are used as the row headings
  • The right operands are used as the column headings
  • The set and/or the operation is placed in the upper left corner
  • The identity element is placed in the first (actual) row and column
• Properties:
  • The first row/column is the same as the headings (reason: identity element)
  • Each element of the set appears once in each row/column (reason: cancellation law)
  • The identity element is distributed symmetrically to the diagonal (reason: inverse element)
  • Associativity has to be checked "by hand"

This Week's Homework

Deadline: December 19, 2019 (Thursday), 19:00.

Format: A4 single page (using both sides is okay; NO cover page), easily readable handwriting (NO printouts), name (kanji and kana) and student number at the top right

Where to submit: Box in front of room O-529 (building O, 5th floor)

Homework 1: Create a Cayley table of the symmetric group of order 3. Use lexical order for the permutations. Use the notation introduced in this lecture.

Homework 2: If we define isomorphic groups as being "the same", there are two different groups of size 4. Give an example of each group as a Cayley table. Hint: Check all the conditions (axioms) for a group. There will be a deduction if you use the same elements of the group as another student.

Glossary

algebraic structure

代数系

group

群 (ぐん)

group theory

群論

inverse element

逆元

inverse, reciprocal

逆数

symmetric group

対称群

closure

閉性

Cayley table

積表、乗積表

multiplication table

九九 (表)

isomorphic

同形の、同型の

group isomorphism

群同形

row heading

行見出し

column heading

列見出し

lexical (or lexicographic(al)) order

辞書式順序