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29 Cards in this Set
- Front
- Back
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Define a Binary Operation *
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Let S be a non-empty set. A binary operation on S is a rule for combining two
elements of S to produce another element of S; given a, b 'in' S we write a b for the element produced by combining a with b. |
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Important things to remember for a Binary Operation *
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(i) a*b must be uniquely de fined for all a,b in S;
(ii) a*b must itself be an element of S for all a,b in S. |
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* is called commutative if
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For all a,b 'in' S.
a*b = b*a |
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* is called associative if
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For all a,b and c 'in' S.
a*(b*c) = (a*b)*c |
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* has an identity if
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an element e exists such that
For all a 'in' S, (a*e = a = e*a) |
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If a binary operation has an identity, it is unique.
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If * is a binary operation and e1 and e2 are both identities, then e1*e2 = e2 as e1 is an identity. And e2*e1 =e1 as e2 is an identity. So e1=e2
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The element of b is called an INVERSE of a if...
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Let the binary operation * on the set S have an identity e, and take a 'in' S. The element b is called an inverse of a if a*b = e = b*a; if such an element exists, we
say that a has an inverse. |
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Define: A Group
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A group is a set G with a binary operation * such that the following conditions are satis ed:
(i) * is associative (ii) * has an identity (iii) every element of G has an inverse |
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Define: Abelian
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A group whose binary operation is commutative is called COMMUTATIVE or ABELIAN
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Proposition 1.3.1.
If G is a group, then each element of G has a unique inverse |
Suppose that b and c are inverses of a; then we have
b = be = b(ac) = (ba)c = ec = c. |
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Proposition 1.3.2.
If G is a group and a,b 'in' G, then the equation ax = b has the unique solution x = a^-1 b. similarly the equation xa = b has the unique solution x = b a^-1. |
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Proposition 1.3.3.
If G is a group and a,b,c 'in' G with ab = ac or ba = ca, then b = c |
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If G is a group with identity e, and a,b 'in' G with ab = e, then b = a^-1
and a = b^-1. |
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Corollary 1.3.5.
Given an element a of a group G, as x runs through the elements of G, the elements ax are just the elements of G in some order, without repetitions; the same is true of the elements xa. |
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IMPORTANT Proposition 1.3.6.
If a and b are elements of a group G, then (ab)^-1 = (b^-1a^-1) |
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A Permuatation of A is
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a bijection from a to itself
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If f and g are bijections
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then so is f composed g. f o g.
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S_n is closed under composition of functions,
i.e. |
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A map f : X ->Y of sets has a (well-defi ned) inverse map...
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is and only if f is a bijection.
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We call S_n the group of permutations or...
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the symmetric group (on n letters)
As we noted earlier... |S_n|=n! |
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S_3 is a...
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non-abelian group. More generally, S_n is non abelian for any n greater than or equal to 3.
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Definition:
The symmetries of F are... |
the "motions" of the plane which fix F as a whole.
A motion means a transformation of R^2 |
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The group D8
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The symmetry group of the square has eight elements,
e,a,a^2,a^3,b,ba,ba^2,ba^3 Here a is the rotation anticlockwise by pi/2 and b is the reflection in the vertical axis of symmetry. These elements satisfy: (i) a^4 = e (ii) b^2 =e (iii) (a^i)b =ba^-i for any i. |
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The Group D2n
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The symmetry group of the regular n-gon has 2n elements: e,a,...,a^n-1,b,ba,...,ba^n-1.
Here a is the rotation anticlockwise by 2pi/n and b is the reflection in the vertical axis of symmetry. These elements satisfy: (i) a^n=e, (ii)b^2=e (iii) (a^i)b=ba^-i for any i. |
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The Viergruppe what is it?
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the symmetry group of the rectangle.
Here there are four elements: the identity e, the rotation a by pi, the reflection b in the vertical axis and the reflection c in the horizontal axis. |
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The Viergruppe Summary:
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is a group with 4 elements:
e , a , b , c satsifying (i)a^2 = b^2 = c^2 = e (ii)The product of any 2 a,b,c is the third one. i.e. ab=ba=c ac=b etc... |
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Equivalence Relations:
We say that R is REFLEXIVE if... |
For all a 'in' S.
aRa |
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Equivalence Relations:
We say that R is SYMMETRIC if... |
For all a,b 'in' S
aRb IMPLIES bRa |
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Equivalence Relations:
We say that R is TRANSITIVE if... |
For all a,b,c 'in' S
((aRb & bRc) IMPLIES aRc) |
