Algebraic Structures

(代数系)

Discrete Mathematics I

11th lecture, Dec. 9, 2016

http://www.sw.it.aoyama.ac.jp/2016/Math1/lecture11.html

Martin J. Dürst

© 2006-16 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.

Algebraic Structure

Very general view on mathematical objects

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

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

Previous 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 (•; ∀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

• Permutations, when seen as exchanging elements, 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 work 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
• Example 1: (ℝ, +) is isomorphic to (ℝ⁺, ·), with f(x) = a^x (a>1)
• Example 2: Three isomorphic groups

Cayley Tables

• Finite groups are usually described using a Cayley table
• 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 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 only appears once on the (main) diagonal, and is distributed symmetrically to the diagonal (reason: inverse element)
  • Associativity has to be checked "by hand"

This Week's Homework

Deadline: December 15, 2016 (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

multiplication table

九九 (表)

isomorphic

同形の、同型の

group isomorphism

群同形

lexical (or lexicographic(al)) order

辞書式順序