Algebraic Structures

(代数系)

Discrete Mathematics I

11th lecture, Dec. 8, 2017

http://www.sw.it.aoyama.ac.jp/2017/Math1/lecture11.html

Martin J. Dürst

© 2006-17 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

• 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

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.

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 a minimal example or explain why such a combination is impossible.

Hint: There are 16 combinations. Two combinations are impossible. One combination needs a set of four elements for a minimal example. Three combinations need a set of two elements for a minimal example. Two combinations need a set of one element for a minimal example. The other combinations need a set of three elements for a minimal example.

Hint: Use {a, b, c} for a set with three elements.

Hint: Present the 16 combinations in a table similar to the tables used in the homework of lecture 4.

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
• Here:
  
  • 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:
  • Associativity (∀b,c,d∈A: (b•c)•d = b•(c•d))
  • (Existence of a) identity element e (∃e∈A: ∀b∈A: e•b = b = b•e)
  • (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

Symmetric Groups

• 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: ∃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: ∀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

• 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 idenity element is placed in the first (actual) row and column
  • The set and/or the operation is placed in the upper left corner
• 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 15, 2017 (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.

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

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