北京邮电大学计算机学院 离散数学 数学结构 群论 chap9-3

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Symmetries of the triangle


There are counter-clockwise rotations f2, f3, f1 of the triangle about O through 120º , 240º , 360º (or 0º ) respectively. f1, f2, f3 can be written as the permutations.

Let G = {e, a} be a group of order 2.

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Nonisomorphic groups of order 3

Let G = {e, a, b} be a group of order 3.
1 2 3 1 2 3 1 2 3 g1 , g2 , g3 1 3 2 3 2 1 2 1 3
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Group of symmetries of the triangle
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Theorem 1


Let G be a group. Each element a in G has only one inverse in G. Proof

Let

a' and a" be inverses of a. a' = a'e = a'(aa") = (a'a)a" = ea" = a".
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Abelian – 阿贝尔群

A group G is said to be Abelian if ab = ba for all elements a and b in G.
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Then

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Theorem 2

Let

G be a group and a, b, and c be elements of G. left cancellation – 左消去律


Then

ab = ac implies that b = c ba = ca implies that b = c
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right cancellation – 右消去律

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Theorem 2 - Proof
1 2 3 1 2 3 1 2 3 f1 , f2 , f3 1 2 3 2 3 1 3 1 2
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Symmetries of the triangle
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Group of order 1, 2

If G is a group of order 1, then

G = {e}, and ee = e. The blank can be filled in by e or by a?

Then

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Theorem 3 - Proof

(a-1)-1 = a

a-1a = aa-1 = e the inverse of an element is unique, So, (a-1)-1 = a (ab)(b-la-1) = a(b(b-la-1)) = a((bb-l) a-1) = a(ea-1) = aa-1 = e similarly, (b-la-1)(ab) = e so (ab)-1 = b-1a-1

(a*b)*c = a*(b*c). a unique element e in G such that

a*e = e*a

an element a' G, called an inverse of a and written as a1, such that

a*a' = a'*a = e or aa' = a'a = e or aa-1 = a-1a = e

a*a' = a*4/a = a(4/a)/2 = 2 = (4/a)(a)/2 = (4/a)*a = a' *a. a*b = ab/2 = ba/2 = b*a

Abelian


So, G is an Abelian group.
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Groups of order 4

Let G = {e, a, b, c} be a group of order 4
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Example 4 - Solution

* is a binary operation


If a and b are elements of G, then ab/2 is a nonzero real number and hence is in G. (a*b)*c = (ab/2)*c = (ab)c/4 a*(b*c) = a*(bc/2) = a(bc)/4= (ab)c/ 4. * is associative.
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Example 4

Let

G be the set of all nonzero real numbers and a*b = ab/2. (G, *) is an Abelian group.

Show

Then


Proof is omitted
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Finite group – 有限群


If G is a group that has a finite number of elements, G is said to be a finite group, and the order(阶) of G is the number of elements |G| in G. A finite group can be represented in the form of the multiplication table.
1 2 3 1 2 3 1 2 3 f1 , f , f 2 2 3 1 3 3 1 2 1 2 3
1 2 3 1 2 3 1 2 3 g1 , g , g 2 3 2 1 3 2 1 3 3 1 3 2
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(ab)-1 = b-1a-1


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Theorem 4

Let

G be a group, and a and b be elements of G The equation ax = b has a unique solution in G. The equation ya = b has a unique solution in G.
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Right cancellation

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Theorem 3

Let

G be a group and a and b be elements of G. (a-1)-1 = a. (ab)-1 = b-1a-1
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associativity



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Example 4 - Solution

2 is the identity

a*2 = (a)(2)/2 = a = (2)(a)/2 = 2*a.

a' = 4/a is an inverse of a
g3
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Left cancellation
源自文库


Suppose that ab = ac. a-1(ab) = a-1(ac) (a-1a)b = (a-1a)c by associativity eb = ec by the definition of an inverse b=c by definition of an identity. The proof is similar to above.


Three additional symmetries of the triangle are g1, g2, and g3, by reflecting about the lines l1, l2, and l3, respectively. Denote these reflections as the following permutations:
Groups – 群
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science & Technology
Beijing University of Posts & Telecommunications
Definition

A group (G, *) is a set together with a binary operation * on G such that, for any elements a, b, and c in G
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An important group


Given the equilateral triangle with vertices l, 2, and 3 Consider it’s symmetries.
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* f1 f2 f3 g1 g2
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f1 f1 f2 f3 g1 g2 g3
Let S3 = {f1, f2, f3, g1, g2, g3} and introduce the operation *, followed by, on the set S3
f2 f2 f1 g2 g3 g1 f3 f3 f1 f2 g3 g1 g2 g1 g1 g2 g2 g1 g3 f2 f1 f3 g3 g3 g1 f f2 f1 f3 g3 g2 f1 f3 f2 g2
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Example 5

Let B = {0, l }, and let + be the operation defined on B as follows:

Then B is a group.
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