Download Definition 1 An AS1 system is a set, say S, with an operation S ! S

De…nition 1 An AS1 system is a set, say S, with an operation S S ! S
where, using juxtaposition to indicate the operation, we have the follow axiom
holding:
1) Associativity: (ab)c = a(bc) for all elements a; b; c chosen from the set S:
2) There exists a special identity element e such that ae = a and ea = a for
any a 2 S.
3) For each a 2 S there exists an element a 1 such that aa 1 = e and also
1
a a = e:
Proposition 2 If af = a and f a = a for any a 2 S then in fact f = e:
Proposition 3 If ab = e and also ba = e for all a then in fact b = a
1
:
Proposition 4 If S has only 3 members then it is commutative (ab = ba for
all a and b).
Example 5 If we let S = f1; 1g and use the ordinary notion of multiplication,
then we have an AS1 system.
Example 6 If we let S be the family of all invertible n n matrices and let the
operation be matrix multiplication, then we have an AS1 system.
Example 7 If we let S be the integers and use addition (!!) as our operation
then we have an AS1. Notice that in this case we might just use the symbol
+ for the operation in which case the multiplicative identity would read as an
additive identity and we denote is none other than 0. Als0, we should write
" a" instead of "a 1 " and na instead of an which looks better with additive
notation since, for example 3a = a + a + a: (If we stuck with multiplicative
notation it would read as a3 = aaa):
Example 8 Let S be the family of all bijections of the set f1; 2; 3; 4g. For
example, one member of S is the bijection indicated below:
1
2
3
4
7!
7!
7!
7!
2
3
4
1
If f and g are two such bijections then by f g we shall mean the composition of
applying g and then f . The result would be a bijection so f g is in our system.
If we take composition to be the meaning of multiplication in this family of
bijections then it becomes and AS1 system.
If S is an AS1 system and H is a subset with the property that ab is in H
whenever a and b are in H then H itself becomes and AS1 system and we say
that H is a subsystem of S.
Notice that an AS1 might only have one element which would have to be
the identity e: For any AS1 system S we can trivially take both feg and S itself
as subsystems.
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Problem 9 Determine whether the systems described are AS1. If they are not,
point out which of the axioms fail to hold.
(a) S = set of all integers, a b = a b (the abstract operation is the usual
subtraction operation).
(b) S = set of all positive integers, a b = ab, usual product of integers.
(d) S = set of all rational numbers with odd denominators, and multiplication de…ned by a b = a + b, ( the usual addition of rational numbers.)
Problem 10 If S is an AS1 such that (ab)2 = a2 b2 for all a; b 2 S, show that
S must actually be commutative.
Problem 11 If the AS1 system S has only 4 elements, show it must be commutative.
Problem 12 Find an AS1 system with exactly 6 elements that is not commutative.
De…nition 13 Let S and T be AS1 systems. A map f : S ! T is called an AS1
morphism if f (ab) = f (a)f (b) for all a; b 2 S. Notice that the multiplication on
the left is the multiplication in S and that on the right is multiplication in T .
Example 14 Let S be the additive system of real numbers (so the abstract multiplication is ordinary addition). Let T be the multiplicative system of positive
real numbers (so the abstract multiplication is ordinary multiplication). Show
that the function f (x) = ex is an AS1 morphism.
De…nition 15 If f : S ! T a morphism, then the subset K(f )
S de…ned by
K(f ) = fx 2 S : f (x) = eg
is called the "kernel" of f .
Proposition 16 The kernel of a morphism f : S ! T is an AS1 system in its
own right if we use the multiplication available from S. (Since K(f ) is a subset
of S we know what it means to multiply elements).
When a subset of an AS1 system is itself an AS1 system with the inherited
multiplication, we call it a "subsystem". So the kernel of a morphism is always
a subsystem.
Proposition 17 If f : S ! T a morphism, then f is an injection (1 to 1) if
and only if the kernel is trivial, i.e. if K(f ) consists only of the identity element
in S:
For example, since ex = 1 only has the solution 0 the kernel of the morphism
in the example above is just the set f0g and we conclude that exponentiation is
1 to 1.
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Problem 18 Consider the AS1 system of invertible 3 by 3 matrices. Denote
it by GL(3). If a matrix A in GL(3) has the property that AT A = I we call it
an orthogonal matrix. The set of all 3 by 3 orthogonal matrices is denoted by
O(3). Show that the subset O(3) GL(3) is an AS1 subsystem.
Problem 19 Suppose S is an AS1 system. Pick an element a 2 S. Let us
de…ne a map S ! S as follows
f (x) = a
1
xa for x 2 S
Show that this map is a morphism. Describe its kernel in terms of some commutation property.
Problem 20 If H is a subsystem of S then we can use it to de…ne an relation
on the set S by stipulating that a b if and only if ab 1 2 H. Show that this
relation is an equivalence relation.
Example 21 The set Z of all integers is an AS1 system if we use addition. (In
this case we write " n" instead of "n 1 "). The subset of even integers, 2Z is a
subsystem of Z. Then the equivalence relation this gives is just that two integers
are equivalent if they di¤ er by an even number. Also, the set of all multiples of
12 is a subsystem and the this gives is just that two integers are equivalent if
they di¤ er by a multiple of 12. We write 12Z to indicate the set of all multiples
of 12. It is a subsystem of the additive system Z.
OK, lets go back the abstract case: H is a subsystem of S. The equivalence
classes we get using the relation de…ned in the above problem are called cosets.
Notice that H itself is a coset since it is just the set of all things equivalents to
e.
The set of all cosets is denoted G=H. So H is just one member of the family
G=H:
So for example for the case of 2Z Z mentioned above, there is only two
cosets: the set of even integers and the set of odd integers. What are the
equivalence classes for the case of 12Z Z.
Problem 22 Let the set S = fe; a; b; cg be turned into an AS1 system according
the the following multiplication table.
e
a
b
c
e
e
a
b
c
a
a
e
c
c
b
b
c
e
a
c
c
b
a
e
Show that the set H = fe; cg is a subsystem. What are the cosets in this
case?
De…nition 23 Suppose that for some AS1 system G we have some element a so
that e; a; a2 ; :::; an 1 are all distinct and that an = e. Then the set fe; a; a2 ; :::; an 1 g
is a subsystem called a cyclic subsystem. If actually fe; a; a2 ; :::; an 1 g = G then
G is called a cyclic AS1 system and the element a is called a generator.
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Problem 24 Show that if G = fe; a; a2 ; :::; a5 g is a cyclic AS1 system then
H = fe; a2 ; a4 g is a cyclic subsystem of G. What is a generator for H?
Problem 25 What are the cosets for G and H as in the last problem?
Now lets set up some more things. Suppose that f : G ! S is a morphism
from some AS1 system to another. We know that K(f ) is a subsystem of G.
Now we want to consider a special subset of S. Let I(f ) = fs 2 S : s = f (g)
for some g 2 Gg: In other words, I(f ) is just the set of all elements in S of the
form f (g) as g varies over all g 2 G: It is just the range of the map f . It is
often also denoted f (G) which makes some sense. So keep in mind that I(f ) is
a subset of S.
Problem 26 Show that if G and S are AS1 systems and f : G ! S is a
morphism then I(f ) is a subsystem of S. We call it the image of f , or the
image of G under f .
Example 27 (and problem) The map f : fe; a; a2 ; :::; a5 g ! fe; a2 ; a4 g given
by x 7! x4 might be a morphism. Is it? What is I(f )?
Example 28 Let O(3) be the AS1 system of orthogonal 3 by 3 matrices. If we
de…ne a map from O(3) to the multiplicative group of nonzero real numbers
as follows
(A) = det(A)
then
is a morphism. Find the kernel K( ) and the image I( ). We will
denote the multiplicative group of nonzero real numbers as R .
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