Algebraic Structures


Discrete Mathematics I

11th lecture, Dec. 8, 2017

Martin J. Dürst


© 2006-17 Martin J. Dürst Aoyama Gakuin University

Today's Schedule


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:


Previously Encountered Examples


Example of Algebraic Structure: Group


The Integers with Addition as a Group (ℤ, +)


The Reals with Multiplication as a Group (ℝ-{0}, ·)


The Positive Reals with Multiplication as a Group (ℝ+, ·)




Permutations as Exchanges


Composition of Permutations


Symmetric Groups


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)


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')


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)


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


Examples of Isomorphic Groups

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


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.



algebraic structure
group theory
inverse element
inverse, reciprocal
symmetric group
Cayley table
multiplication table
九九 (表)
group isomorphism
row heading
column heading
lexical (or lexicographic(al)) order