# Algebraic Structures

(代数系)

## Discrete Mathematics I

### 11th lecture, Dec. 8, 2017

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

# 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: xA:xRx; ∀xA: (x, x) ∈ R
2. Symmetric: ∀x, yA: xRyyRx;
x, yA: (x, y) ∈ R ⇔ (y, x) ∈ R
3. Antisymmetric: ∀x, yA: xRyyRxx=y
4. Transitive: ∀x, y, zA: xRyyRzxRz

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:
• 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,cA: bcA)
• Three axioms:
• Associativity (∀b,c,dA: (bc)•d = b•(cd))
• (Existence of a) identity element e (∃eA: ∀bA: eb = b = be)
• (Existence of an) inverse element b' (∀bA: ∃b'A: bb' = 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·cd = 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·cd = 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: ∃eA: ∀bA: eb = b = be

Theorem: The identity element of a group is unique
(∃cA: ∀xA: cx = x) ⇒ c = e

Another way to express this: There is only one identity element
(|{c|cA, ∀xA: cx = x)}|= 1)

Proof:

cx = x [inverse axiom, closure]

(cx)•x' = xx' [associativity axiom]

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

ce = e [identity axiom]

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

# Group Theorem: Uniqueness of Inverse

Existence of an inverse: ∀bA: ∃b'A: bb' = e = b'•b

Theorem: Each inverse is unique
a, b∈A: (ab = eb=a')

Proof:

ab = e [applying a'• on the left]

a'•(ab) = a'•e [associativity axiom]

(a'•a)•b = a'•e [inverse axiom]

eb = 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: (ac = bca=b)

Proof:

ac = bc [applying c' on the right]

(ac)•c' = (bc)•c' [associativity]

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

ae = be [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:
• hH: ∃gG: h = f(g)
• g1, g2G:g1g2f(g1)≠f(g2)
• g1, g2G: f(g1g2) = 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) = ax (a>1)
• Example 2: Three isomorphic groups (shown as Cayley tables)
e a b G e a b a b e b e a
0 2 1 K 0 2 1 2 1 0 1 0 2
0 1 2 H 0 1 2 1 2 0 2 0 1

# 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

group
group theory

inverse element

inverse, reciprocal

symmetric group

closure

Cayley table

multiplication table

isomorphic

group isomorphism