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:
- Reflexive:
∀x∈A:xRx;
∀x∈A: (x, x) ∈
R
- Symmetric: ∀x, y ∈A: xRy
⇔ yRx;
∀x, y ∈A: (x, y)
∈ R ⇔ (y, x) ∈ R
- Antisymmetric: ∀x, y ∈A:
xRy ∧ yRx ⇒ x=y
- 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:
- One permutation that keeps the same order: (1, 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
| G |
e |
a |
b |
| e |
e |
a |
b |
| a |
a |
b |
e |
| b |
b |
e |
a |
| K |
0 |
2 |
1 |
| 0 |
0 |
2 |
1 |
| 2 |
2 |
1 |
0 |
| 1 |
1 |
0 |
2 |
| H |
0 |
1 |
2 |
| 0 |
0 |
1 |
2 |
| 1 |
1 |
2 |
0 |
| 2 |
2 |
0 |
1 |
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
