(代数系)

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

© 2006-17 Martin J. Dürst Aoyama Gakuin University

- 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

Hasse diagrams, equivalence relations and order relations in matrix representation

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`⇒`xR``z`

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*.

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.

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

- 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

- 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})

- Associativity
(∀
- Note: Commutativity is not necessary

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

- 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})

- 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`

- 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 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)

- 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

- 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`

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)

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)

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)

- Two groups (
`G`, •) and (`H`, ∘) are`isomorphic`if there is a function`f`so that:- ∀
`h`∈`H`: ∃`g`∈`G`:`h`=`f`(`g`) - ∀
`g`_{1},`g`_{2}∈`G`:`g`_{1}≠`g`_{2}→`f`(`g`_{1})≠`f`(`g`_{2}) - ∀
`g`_{1},`g`_{2}∈`G`:`f`(`g`_{1}•`g`_{2}) =`f`(`g`_{1})∘`f`(`g`_{2})

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

- They have the same number of elements
(|

- Example 1: (ℝ, +) is isomorphic to (ℝ
^{+}, ·), with`f`(x) =`a`^{x}(`a`>1) - Example 2: Three isomorphic groups (shown as
*Cayley table*s)

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 |

- 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"

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.

- 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