SOME TOPICS IN THE REPRESENTATION THEORY OF FINITE GROUPS AND
LINEAR ASSOCIATIVE ALGEBRAS
HONORS
THESIS ID 499
BALL STATE UNIVERSITY
ADVISOR:
MR.
RANDAL P. MILLER
META DARLENE KEIHN
JULY 15t 1966
·
-
,..+
I
1'.·/
I
."
Acknowledgements
Since mathematical notation and definition are still somewhat a
matter of individual whim. I have chosen to use that which Mr. Miller
useS in his lectures.
I am therefore very grateful to Miss Paula
Lovelace and Miss Rebecca Russell for the use of their classnotes
taken in Mathematics
l t ll
and Mathematics 512.
I am particularly in-
debted to Mr. Miller himself for his help and guidance.
INDEX
Introduction •••••••••••••••••••••••••••••••••••••••••••• 1
I.
II.
III.
IV.
v.
Basic Definitions ••••••••••••••••••••••••••••••••••• 1
Cayley's Theorem ••••••••••••••••••••••••••••••••••• 7
Representations of Groups •••••••••••••••••••· •••••• 9
Linear Algebras ••••••••••••••••••••••••••••••••••• 15
Conclusion ••••••••••••••••••••••••••••••••••••••••• 22
Bibliography ••••••••••••••••••••••••••••••••••••••••••• 23
Some Topics in the Representation Theory of Finite Groups
and Linear Associative Algebras
Introduction
For many years algebra was considered to be merely symbolic
arithmetic; a short cut for easier problem solution.
mathematicians became aware that
sO:r~e
Eventually
of the symbolic statements from
this generalized arithmetic, which were true when numerals were plugged
in, were also true when the symbols were replaced by other objects.
'rhus algebra evolved into the study of mathematical systems.
One of the
most fundamental of the systems with "nice' properties is the group.
To
a large extent this paper will illustrate some methods of analyzing
abstract algebraic systems which have been devised by mathematicians.
In particular after basic definitions have been given, the emphasis
shall be upon representing groups by other groups to facilitate that
analysis.
However, as abstract as group theory may appear, it must
not be forgotten that group theory does have important applications in
physics and chemistry.
I.
Basic Definitions
In any branch of mathematics one must begin with some undefined
terms which are intuitively clear, and build upon these logically to
develop some ideas by means of definitions, postulates, and theorems.
Our only undefined term necessary for this paper is set.
A set is a
basic formally undefined noun--intuitively a collection of objects-to which we assign a basic formally undefined predicate, the elementhood predicate.
Intuitively, an object either is or is not an element
1
2
of a set.
The term subset is then formally defined as follows:
Definition 1.
A set T is a subset of a set S if and only if every
element of T is also an element of S.
A set T is a proper subset of a
set S if and only if T is a subset of S, but S is not a subset of T.
Other definitions essential for a complete hierachy of logically
developed concepts include:
Definition 2.
Let a and b be sets.
(a,b)
a.
Remark:
=
b
(denoted
((a},(a,b}}.
=c
a
Let a and b be sets.
a x b
and read
a x b
Definition 4.
=
and
(x,y)
subset of the Cartesian product
b
= d.
By the Cartesian product of a and
tla cross b")
Let a and b be sets.
Definition 5.
we mean
This definition implies that (a,b) and (c,d) are
equal if and only if
Definition 3.
By the ordered pair (a,b),
I
"Ie mean the set
(x c a) 1\ (y c b)}.
A relation from a to b is any
a x b.
Let a and b be sets, and R be a relation from a to b.
R is called a functional relation if and only if no two different pairs
with the same first entries are elenents of R.
Definition 6.
Let X and Y be sets.
if and only if:
and
Then f is a function from X to Y
1)
f is a functional relation from X to Y;
2)
the domain of f is all of X.
That is, the set of
all the different first entries of the various ordered pairs in f is all
of X.
a.
Notation:
The set of all functions from a set X to a set Y
is denoted by yX.
X
to Y.
Thus
f c yX
means f is a function from
A manning is a function.
Definition 7.
If
a
called a mapping of S
such that
Definition
ax
8.
is a mapping of a set S into a set T, then
~
T if for each
yeT, there is some
a
is
xeS
= y.
A mapping
a
is called a
1:1 mapping or injection of
S into T if and only if any two distinct elements of S have distinct
images in T.
Definition 9.
A permutation on a set S is a
1:1 mapping of S onto it-
self.
Let S be a set and let R be a relation from S to S.
R
c: S
x S.
That is,
Then certain properties of R may be defined as follows:
Definition 10.
for every x c S, then
R
is said to be reflexive
iff
~
is said to be symmetric
iff whenever
(x,x) c R.
Definition 11.
(x,y) c R, then
(y,x) c R.
Definition 12.
R is said to be transitive
(y, z) c R, then
Definition 13.
(:x,y) c H, and
(x,z) c R.
Let E be a relation from a set S to S.
equivalence relation
Definition 14.
iff whenever
iff
E is called an
E is reflexive, symmetric, and transitive.
A mathematical system S is a set
S
=
(E, 0, A)
where
E is a nonempty set of elements, 0 is a set of relations and operations
on E, and A is a set of axioms concerning the elements of E and O.
Having accepted these fourteen definitons, we are now ready to
define what is meant by a group.
Definition 15.
Let G be a nonempty set (furnished with an ,equivalence
relation) and let
from G x G to G).
o
0
be a binary operation on G (i.e.,
0
is a function
The mathematical system (G, c ) is called a group
has the following properties:
iff
4
1)
for every three elements x,y,z of G,
is associative:
0
x
0
e
There is an element
2)
e .. x
L
3)
= x.
For every element
-1
L
c G
(x
y)..
0
z.
so that for every
x c G,
(Left identity element)
x c G, there corresponds an element
inv4~rse.)
(Left
x = e.
so that
c G
xL
=
(y ., z)
Of course there are other similar definitions possible, but all
are equivalent to the above.
amples?
Given this definition, what are some ex-
Integers under usual addition, rational numbers under usual
addition, nonzero rational numbers under multiplication, and complex
numbers under usual addition name a few well-known groUi)s.
look at arithmetic modulo
five.
The addition table below shows speci-
fically:
0
1
234
001
234
1
1
2
340
2
2
340
3
3
4
+
Suppose we
<3
2 +
a + (b + c) = (a + b) + c
in general,
where a, b, and c
the table.
+ 1) = (2 + 3) + 1;
can be any €,lements of
The additive identity is 0, and
1
0-0; 1-4;
each element has an inverse:
012
2-3; 3-2; and 4-1.
440
1
2
Thus we must again
3
have a group.
Theorem 1.
=
0
a -1
L
(a,
-1 c G
~
Froof:
v
If
= e.
v .. e.
=v
is a group and
)
~nplies
a
Tien
-1
~
Ct
-1
0
~
c;
a -1
l
= e.
that there exists a
= (e" a) ..
Therefore
is also a right identity, consider:
e "a
= a.
a c G, then
-1
~
=
(v
a. a
ao
-- e, and
"v" c G such that
0
-1
L
-1)
~
= e.
c-
ae
-1
~
=
v .. ( -1
~
To see that e
.
L
aoe
Therefore e is the identity element.
no need of the subscript L and it can be dropped.
Note that we now have
a
)
-1
"~
5
There is a group-theoretic analogue for the set-theoretic concept of subset.
Definition 16.
If
G.
0
This is the idea of subgroup, whose definition follows:
Let (G,O) be a group and let H be a nonempty subset of
is the restriction of ()
H
to H, (i. e.,
n ,,),
"H =(H x H x G)
then the system (H, C)H) is said to be a subgroup of (G, c) if (H,
C
H) is
a group in its own right.
We denote the statement that
H< G.
notation
&
is a subgroup of (G,o) by the
It can be shown that (H,oH) is a subgroup of (G,o) when
H is a nonempty subset of G iff
both a • b
(Ht~H)
Hand
-1
a
&
1) for every element a & Hand b & H,
H; or
if a
2)
&
Hand b
&
li, then a., b
-1
&
H.
Of course not every subset of G will be a subgroup under the defined
operation.
Definition 17.
Let (G,o) be a group.
The order of G, denoted
is defined to be the cardinal number of the set G.
finite
iff
(G: e)
[G:e],
(G,o) is said to be
is finite.
Examples:
The order of the group
(Z,+) of integers under addition is
infinite.
The order of the group
Z
Definition 18.
G.
n
of integers modulo n is n.
Let (G,o) be a group and let S be a nonempty subset of
Then S is called a complex of G.
If S is a complex'of the group (G,d), then the smallest subgroup
(H, eli) of G which contains S as a subset is called the subgroup generated
~
2'
notated by
Definition 19.
(S).
(S)
= (S,
'S)
Let (G,o) be a group.
there exists some element g & G so that
Example:
+
0
1
001
110
iff
S
< G.
(G,e) is said to be cyclic
G
iff
= ({g}).
This table gives a cyclic group of order
two whose generator is 1.
6
It is well-known that every subgroup of a cyclic group is (:yclic.
Definition 20.
If H is a subgroup of (G,o), then the set
for a fixed element a of G, is a left coset of H.
a representatitive of aH.
aH
=
l
{a·h , h cH},
The element a is called
Right cosets are similarly defined.
Example:
.
e
a
b
c
d
f
e
e
a
b
c
d
f
a
a
b
e
f
c
d
•
e
a
b
b
b
e
a
d
f
c
e
e
a
b
c
c
d
f
e
a
b
a
abe
d
d
f
c
b
e
a
b
b
f
f
c
d
a
b
e
A subgroup of this group would be:
e
a
If we let a, b, or e be the representative, the left cosets are:
eH = aH = bH = H = {e, a, b).
If c, d, or f is the representative, then we have the left cosets:
= dH
cH
= fH
= {c,
Note also that the set
whose generator is
(G,.) and
c.
d, f).
A
= {e,
O}
is also a cyclic subgroup of G
This example implies that if H is a subgroup of
aH and bH are cosets of H then
1) aH
=
bH
or
2)
aH and bH
are disjoint.
Definition 21.
The Gubgroup H of the group G is normal in G if and only
if for every x in G the left coset xli equals the right coset Hx,
x -1 Hx
~
H.
H, a normal subgroup of G, is denoted by
H
6
or
G.
, ° ) and (G 2 , °2 ) be groups, and let
c G~l.
l 1
to G .) tp is said to be a homomorphism
(i.e., <fJ is a mapping from G
l
2
Definition 22.
iff
(x °1 y)
Let
4J ::
(x
cp
(G
tP)
0
2 (y
IDeskins, Abstract Algebra,
<1»
for every
pp. 212-213.
x c G
l
and every
y c G •
l
7
is called a mono;norphism.
1:1
A homomorphism which is
A homomorphism which is onto is called an eEimorEhism.
A homomorphism which is both a monomorphism and an epimorphism is
called an isomorEhism.
II.
Cayley's Theorem
If we have a set S, as we said before, a permutation of S is any
1:1
onto function from S to S.
possible
permutation~
123\
(
1 2
3)'
(1 2
Let
S
= (1,
3}.
2,
3)
\ 1 3 2,
(1 2
3)
(1 2
3 2 1
3)
2 1 3
describe the above are respectively:
and
(3:)
with appropriate notation, are:
(1 2
,
(1 3 2).
(1),
3)
2 3 1
A cycle is a shorthand way of writing permutations.
(1 2 3),
Then the
(2 3),
1 2
,and
(
3)
3 1 2 •
The cycles which
(1
3),
(1 2),
Note we are using left hand notation, reading
cycles and products of cycles from right to left.
Defintion 23.
A transEosition is a cycle of order two.
Is this set of permutations under operation of composition of
functions a gro1ip?
is
(1),
Composition of functions is associative, the identity
and each permutation has an inverse--the first four are their
own inverses, and the last two are inverses of ea.ch other.
system is a group.
with
n
In general, the group of all permutations of a set
elements is called the symmetric group on
designated
'rherefore the
n
symbols and is
S •
n
Informally, a representation is a general way of using a homomorphism
from abstract systems (in which the elements are unknown and just the set
is defined) to concrete systems.
If we can exhibit a representation of
abstract groups as groups of permutations, it will be a simple matter to
-----------~~--~--
8
represent permutations as matrices and computation will be greatly simplified.
The theorem which says that abstract groups can be represented as
groups of permutations is known as Cayley's Theorem.
Its formal statement
and proof are as follows:
Theorem 2.
Let G be a group.
(Cayley's Theorem)
to a subgroup of
i.e., every group is
SrG:e~;
Then G is isomorphic
to a group of
isomor~hic
permutations.
Let
Proof:
g
be the permutation in
1)
the range of
(since G is closed under multiplication) and
2)
g
g
Let x and y be elements of G and assume
if they were equal, then xg
3)
g(xg- 1 )
g
is onto.
rn. G ~ S[ G:e J
'f'.
1.
(jJ
g1
1)=
Prove:
Proof:
and
gl;
gl(x)
= g2(xg1 )
g ifJ
yo
Let
g1(/J g2rj),
= xg1
g2CP
q)
= xg g •
1 2
is one-to-one.
gl
g2 (J) = g2'
every
for
a contradiction.
pre-image of x is
g
be given by
= g1 g 2 =
g1 g 2rfJ
:. g2(gl (x»
2.
The
g(x) ,t g(y),
= y,
x
1:1.
xg
-1
since
= g.
'Ile
assert that
Cf
is a
1) ~ is a homomorphism,
is 1:1-
Prove:
Proof:
x c G.
which implies
To prove this, we must show that
monomorphism.
2)
Let
= yg,
x ;l y.
is
g
= (xg-l)g = x(gg-1) = xe = x.
Let
and
which maps
SCG:eJ
really is a permutation of G since
x onto xg.
is G
and let
g & G
x c G.
~
g2
where
x -1( xg )
1
= xg 2 •
= x -1 (xg2 )
= g2;
g2(y)
= yg2·
is a homomorphism.
be elements of G.
g2(x)
¥ZX = x(g1 g 2)·
where
Then
If
g1
c[) = gl'
gl "" g2'
for every
then xg
1
x c G.
so
a contradiction.
where
gl(x)
= xg 2
= x g1 ,
for
By associativity,
but this is
and
~ must be one-to-one.
9
III.
Representations of Groups
Having proven Cayley's Theorem, we are now ready to show that not only
can we represent groups by permutations, but that these permutations can
be represented as matrices.
Definition 24.
What is a permutation matrix?
A permutation matrix is an n x n matrix of
in which there is one and only one
Example:
"1"
lis and
O's
in each row and column.
(1 2 3) is represented
The cycle representation of the permutation
in a permutation matrix as follows:
0)
0
I
(
(remember left hand notation is being used, so read
001
100
Theorem 3:
the cycle from right to left.)
Every permutation can be represented as a matrix.
Our proof will be made clearer if we first prove a lemma.
Lemma 1:
Every permutation in the symmetric group
S
of order
n
is
n!
a product of transpositions.
Proof:
Consider the cycle:
(aI' a 2 , a , ••• , a k ).
3
Observe that
Every
permutation in S
n
Example:
(1
is a product of transpositions.
2 3 4 5)
=
(1 5) (2 5) <3 5) (4 5).
Now since any permutation can be written as a product of cycles and
each cycle is a product of transpositions, then if we exhibit a homomorphism
between permutation matrices and transpositions, this homomorphism is
sufficient for proving permutations in general can be represented by matrices.
Proof:
Let
(i j)
and
(k q)
be two arbitrary transpositions in S •
n
Define a function h from the set of tra.nspositions of S
n
n x n permutation matrices by:
to the set of
10
hei j) = the n x n matrix obtained from the n x n identity matrix In
.th
J
rows.
by the interchanging of the ith and
n
= ') A
+ Aij + Aji
m' = Imm
m ¢ {i,j}
th
is the n x n matrix with 1 in the (r, s)
where
place and O's elsewhere.
To show that h is a homomorphism, we shall show that
q»
h(ei j).(k
Case 1.
{i,j}
n
Case 2.
{i,j}
n {k,q}
{k,q}
(i j) (k q)
Case
3.
=¢
= {j}.
=
{i,j} = {k,q}.
= h(i
For
j)oh(k
q).
definitenes~,
say
q
= j.
Then
(i j) (k j) = (i k j).
For definiteness, say i
(i j) (k q)
= (i
q»
=\
L
=k
and j
=
q, so that
= (1).
j) (i j)
n
Case 1.
h«i j)o(k
Amm + Aij + AJ' i + ~q + Aqk •
m=l
mt{i,j,k,q}
n
and
h«i j» =
Amm + Aij + Aji
>
On the other hand,
m=l
m~{i,j}
n
h«k
q»
= \"
l"1.
Amm + -\q + A qk.
m~{k,q}
Now the matrix basis elements A
are multiplied according to the formula:
rs
ArsAtu
= Aru • cf (st)
=1
if
where
defined by
d(st)
s
Therefore,
h«i j»oh«kq»)
d est )
d est ) = 0
= t,
and
=(.~L=l
Amm + A.~J. + A..
J1
m
+
mt{i,j}
m¢{~q}
m¢{i,j}
if
s F
m=
Amm
J( ~n
m~{i,j)
=("£1~ A_j~ (1n Amm;) (n1 A,.,.)'" ~q
cf
is the Kronecker
-function:
m¢{~q}
n
~l
+( m-
\
A
)
mm",
m¢{i, j} /
+
A •• (
~
.
~J m~l
A
\ +
mm)
m¢{k,q} "
11
A:LJ
.. --kq
A..
+ A:LJ
.. Aqk + AJ:L
.•
C~/
~
+
Amm
.. Aqk
+ AJ:L
A J:L-Kq
•• A..
m=-l
m~{k,q}
n
• I'
A
m=~
+
mm
~q
+ Aqk + Aij + 0 + 0 + Aji + 0 + 0
mF{i,j,k,q)
n
=m~
+ Aij + Aji + ~q + Aqk •
Amm
m,{i,j,k,q)
~
= h«i
h«i j»oh«k q»
j» = h(i
he(i j)o(k
Case 2.
j)o(k q».
k j)
A
mm
n
But
=m~
h«i j»
and
m~(itj)
:.h«i jll.h«k jll -
- ()\"n
m~
A
mm
m;{i,j}
)f j'n
'l
m=~
"
heCk j»
Amm
(i
ml( i, j}
\ (n~
A )\+
mm
(1
Amm + Aij + Aji)
\ m::
A ,\ ~j
mmJ
m¢{i,j)
m¢{j,k}
In
I
Amm +
n
+
~
m~-l
Amm + ~j + Ajk •
m1'{k,j)
'm,t (j ,k}
\
=
n
~j +
Ajk)
,
,m~ Amm,.1
In
Ajk
m~{i,j)
+ A:LJ
..
(!
b
\
. Amm,)'
/
m¥{j,k)
m=
)
+ A. . A.. • + A: . Ajk + A..
A
:LJ -KJ:LJ
J:L ( m="l mm,
m,i( j ,k}
n
= m~
Amm
+ l\j + 0 + 0 + 0 + Aik + Aj i + 0 + 0
m¢,(i,j,k)
n
= m{"l Amm
+ Aik + Aji +
~j.
m;{i,j,k}
:.
Example:
=
h«i j)o(k j»
(2
3)
<3
4)
,
=
(2
h«i
4 3)
j»
0
heCk
j».
(1 000) (
001 0
o 1 0 0
o 001
1 0)
0 0
0 1 0 0
0 0 0 1
001 0
= (;
gg ~)
o 1 0 0
001 0
12
i j )o(i
h«
Case 3.
j» =
n
heel»~
=
or
In
n
But
h«i j»
=~'
m=l
Amm
m~
A
m.m
•
+ Aij + Aji-
m~{i,j}
+
+
A~
Aill
Amm)
m~(i,j}
+
n
- L Amm
+ 0 + 0 + 0 + 0 + Aii + 0 + Ajj + 0
m=l
m~{itj}
n
=)
A
m~
DIm
•
h«i j)·(i j»
= h«i
=
j)}
0
I
_
n
h«i j» •
Therefore every transposition, and thus every permutation, can be
expressed as a permutation matrix.
If a representation is a monomorphism, it is said to faithful.
We assert that
h
above is faithful.
Now, since every two permutations PI
product of transpositions, then
1)
and
PI; P2
can be written as a
if and only if:
one of them (say PI) involves a transposition
is not a factor of the other.
(i j) which
13
or
2)
are the same,
and
if all the transpositions of Pl
there is at least one pair of nondisjoint transpositions
(i j) and (j k)
and
which appear in different orders in
Pl
P2;
provided we write permutations as products of transpositions only in
the way described in Lemma 1 on page 9.
Now since
behave exactly the same in
need only consider
Pl
and P2
tl t2
···tmt m+l ••• t n
will
under a homomorphism h, we
h(i j) and h(k q).
n
h(k q)
and
A
mm
=
2
Amm
m=l
m¢{k,q}
A = A"
Now, by definition of the equality of two n x n matrices,
and only if
= Anrs
In h(i j),
h(i j)
A ..
l..J
I
for every
= 1.
In
h(k q).
(r,s) th
entry in A and An.
h(k q),
Aij
•
I
h(Pl )
Since all the other transpositions except
~
= O.
h(P2)·
(j k) and (i j) are the same
and in the same order, we need consider only
h«i j)Q(j
k»
and
n
Above we saw that
= m~
h«i j)c(j k»
+ Aik + Aji +
Amm
~j.
m~{i,j,k}
n
m~
Similarly we can show
Amm
+ Aij + Ajk +
~i·
mf{i,j,k}
Since
Aik f ~i
1
Aji
1
1
Aij
•
Ajk
# ~j'
h(Pl)
f
h«i j)'(j k»
h(P2)·
f
h«j k) (i
j».
14
Therefore representation of permutations by means of matrices is a
faithful representation.
Although the representation of groups with permutation matrices
has been extensively studied and frequently employed to discover more
about groups, this is by no means the only representation possible.
Definition 25.
N ~ G.
Let G be a group and
The factor group
GIN
is the set of (left) cosets with respect to N multiplied according to
the formula:
xN., yN
= xyN.
GIN is often called the quo-
tient group of G with respect to N.
To show the above definition indeed gives us a group:
1)
the identity element is eN, since
2)
the inverses exist:
3)
.. is associative:
x(yz)N = xN
e
•
xN
0
x -IN = xx -IN
(xli" yN) '" zN
(yzN) = xN
c>
=
= xeN = xN.
XN9 eN
(xyN)
= eN.
0
zN
=
(xy)zN
=
(yN .. zN).
GIN is a group •
Let us consider the Klein 4 group and the factor group of
order two,
0
a
b
c
0
0
a
b
c
a
a
0
c
b
b
b
c
0
a
c
c
b
a
0
Let
h
V4/(a) •
V4/(a)
where (a) = {O, a}.
be a homomorphism from V
4
to
Then
h(gl + g2) = (gl + g2)(a) = glCa) + g2{a)
= hCg l ) + h(g2)·
Let k be a homomorphism from V4/(a) to the group of order two.
k.
Then
h would be a homomorphism (since it is the composition of two
homomorphisms) from V4 to the group of order two.
This is called an
induced representation.
First to show k is a homomorphism, let k«O,a»
k«b,c»
= -1.
= 1 and
15
(O,a)
(b,c)
•
1
-1
(O,a)
(O,a)
(b,c)
1
1
-1
(b,c)
(b,c)
(O,a)
-1
-1
1
o
Quite obviously, since mere sUbstitution gives one table from
k
is in fact an isomorphism.
if
N is the identity group.
We note that
k.h
anothe~,
is faithful if and only
This representation may of course itself
be represented by matrices.
IV.
Linear Algebras
We shall now consider vector spaces and algebras and exhibit some
representations of them.
Definition 26.
A vector space is a system
1)
F is a field
(of scalars);
2)
V is a nonempty set of objects called vectors along with an
(V, +)
operation + on V so that
and
(V, F, .) where:
3)
is a function from
0
F x V to V
is an abelian group;
(called scalar multiplication)
such that:
a)
lCl = Cl
for every
b)
(c c )Cl
l 2
= c l (c 2Cl);
c)
C(Cl + 13) = CCl + C~;
d)
(c
l
+ C )Cl
2
= clCl
Cl
in V;
+ C2 Cl.
Vectors shall be denoted by Greek letters; scalars shall be denoted
by letters of the Roman alphabet.
A well known example is the set of all ordered n-tuples of real
numbers, known as n-dimensional Euclidean space
Rn.
16
Definition 27.
Let F be a field.
a vector space
dY
A linear algebra over the field F is
over F with an additional operation called multiplication
of vectors which associates with e<,ch pair of vectors
tor
67
in
1)
ot(/3 p)
2)
ot(/3 + p)
3)
for each scalar c in F,
Example:
and
called the product of ot
otl3
=
ot, 13
in
ct
a vec-
in such a way that:
13
(ot l3)p;
= al3
+ ap
The set of n x n
and
(a +13)p
c (a(3)
=
+ I3p;
(ca)13
= a(cl3).
matrices over a field, with usual operations,
is a linear algebra with identity.
an algebra with identity
= ap
In particular, the field itself is
(over itself).
We shall now concern ourselves with representing these abstract
"things"
called vectors and algebras as matrices and showing this repre-
sentation to be a homomorphism.
In order to do this we need some basic
definitions.
Definition 28.
Let V be a vector space over F.
A subset S of V is said
to be linearly dependent if there exist distinct vectors
in S, and scalars
c l ' c 2 , ••• ,c n
aI' a , ••• a
2
n
in F, not all of which are zero, such
that
A set which is not linearly dependent is called linearly independent.
Definition 29.
Let V be a vector space.
independent set
d.3
A basis for V is a linearly
of vectors in V which spans V, i.e., every vector
a & V can be expressed as:
i
= 1,
n n
where
2, ••• ,n.
Example 1.
defined by:
En
••• + a 13
= (0,
The standard basis for Rn ,
El
=
(1, 0, 0, ••• ,0);
0, 0, ••• ,1)
E2
consisting of
=
E , E2 , ••• ,E
l
(0, 1, 0, ••• ,0);
n
•••
is a well-known example.
Example 2. In the example of a matrix algebra, the basis elements would
be of the form:
l3 ij
=n
x n
matrix with 1 in the (i,j)th place and
17
n
O's elsewhere.
I
Then
L'
(the identity matrix) =
n
~i·].
and any
t
i=l
other n x n matrix
(~l
\
a~l
a 12 •••
\
\a~l
al~
•
•
•
•
•
•
I
n
n
a 22 ••• a 2n
2
=I
aij~ij·
1=1 j=l
f
I
a
'
a
n2 ••• nnJ
Therefore this definition does give a generating set which is clearly
independent.
Definition 30.
space
W(F)
a
Notation:
V(F),
V(F)
is a vector space homomorphism from
(aa + b~)A = a(a)A + b(~)A,
and only if:
and any
The mapping A from vector space
and
f)
to a vector
V(F)
to
W(F)
for any a and b in F
in V.
If we have a homomorphism A as defined above from
we shall denote this statement by
A
C HomF(V,V),
a vector space homomorphism from the vector space
to itself over
F.
Definition 31.
Let
if
6r 1
and
V
V(F) to
read
"A is
over the field
be algebras over the field
F.
F
An
algebra homomorphism is a vector space homomorphism which is also a ring
homomorphism, that is , it also preserves
Definition 32.
mUltiplication.
The dimension of a vector space
is the number of elements in a basis
of V(F).
V(F), denoted
V(F)
dim(V),
is finite-dimen-
sional if a basis contains only a finite number of vectors.
Definition 33.
The dimension
2!
~ algebra
6t
(as an algebra)
equals
its dimension as a vector space.
Theorem 4.
If
dim(V)
= n,
of a finite vector space
Proof;
Case 1.
68
2
(i. e., there are n elements in a basis
d3 1
V(F) ), then every basis of V(F) has n elements.
equals a set of vectors with more than n vectors.
18
\-Je
If
shall prove that
dim(d3 )
2
= x,
Ci:>2
x
=n
is linearly dependent.
13*i c
then each
+ 1,
~i c
can be expressed in terms of
<8 1 ,
113 2'
i c {l, 2, ••• ,x}
i c {l, 2, ••• t n} •
which implies
- al-la 13
;
n n
which implies
13*2 =
- b- l b (:3 ;
2 n n
•
•
•
•
•
•
which implies
+ n2132 + ••• + nn(:3n
-1
-1
*
* •
n n 13*n - n n nl131 - ••• - nn-1 n n-l f3 n-l
13;
Then
x = n + 1)
(where
+ x
i.e.,
But from the above implications,
(:3 •
n n
Therefore
of the vectors in
basis.
Therefore
Case 2.
63 3
l3 i c
011
CB 2'
so
Ci3 2
(:3*
x
can be expressed in terms
is linearly dependent and thus not a
dim(V) ~ n.
has m vectors,
m
=n
- 1.
have n equations in n - lnvariables,
> ai~i =
we get
Assume it is a basis.
13i'
Can be expressed in terms of
ous equations)
f3 i 's,
can also be expressed in terms of
where
13i c
633 -
by elementary algebra.
O.
But since
CB
1
Then
Since we
(simultane-
is a basis, the
i=l
vectors are linearly independent and this is a contradiction.
08 3 is not a basis.
Thus
dim(V)
1
n.
Therefore
Therefore the dimension of a
vector space is unique.
Definition 34.
Let
~
be a finite dimensional linear associative alge-
bra over a field F.
Let V be a finite dimensional vector space over F.
A representation of
Or is an §l:Lgebra homomorphism from
~
to
HomF(V, V).
19
Theorem
6t
5.
Let
d(
be an
n
dimensional algebra over the field F.
has a faithful repreGcntation in
c9r
bra monomoq,hism from
Proof:
into
c:9t
Case 1.
HomF(V,V);
to ;rk (F)
tV
i3
2
i.e., there is an alge-
where k is to be specified.
has an identity.
117 n (F).
Then
eft
In this case map
, ••• , 'f. }
n
faithfully
ctr
be a basis for
with mul ti-
plication of the basis defined by:
'\J
N
~i' Pj
:>efine
(/J : 6B ...
*n
the matrix of structural constants
(r,s)
whose
th
n
7J( rs -)
- s=l
=
and ('n(~.)
'f'
~
by mapping,
(F)
7Jt
i.e., the
(ex.),
r~s
entry is the structural constants
ex
p
ris rs'
an algebra monomorphism.
n x n
matrix
ex . •
r~s
n
L
r:':l
where ~ is
rs
rs
{J
the linear extension of
to all of
1J a
This makes the extension of
cfI
is
vector space
homomorphism.
'~le
(jJ
must no'", prove
is an algebra homomorphism.
'fo verify this
assertion, it will be sufficient to prove that:
a~.. ~ .) 'f"/()
J
By definition,
= (~.)
rf) • (r; , ) if)
~ 'r
J
(~., ~.)
~
J
=
(matrix multiplication).
()~'
a. 'k(;k)
~k=l
= )-
~J
!
.~
ex. 'k a k r .
~J r s rs
k=l s=i ~
On the other hand:
(~.) q} (~,)J qJ
=(
1
=
~e
=
~
f
n
n
k~l r=l
I
~
ex
p
rik rk
n
I
X~ s~
n
m~l
Cl r ikClk'JS fl rs •
r=l 5=1 k=l
n
now need to conclude the equality of (1) and (2) above.
n
)
a ikCl, ,
k'=l r
KJS
tivity of
qJ
But
) a .. kex k
k:1 ~J
r s
( this relation will be shown to follow from the associa-
the multiplication of basis elements of
~
);
that is if:
20
=
a
=
n
n
)
\'
'>
p=~
)
Ll
q=
=
a .. a k
~Jq q P
\
p=
q=.L.
L.,
\
a
L a.~J·k r k s
k=l
1
Finally
Let
A
= )
k=l
is
element of
A
t
Using this it is evident that
"
E:
&t
Therefore
a ikak . •
r
JS
rp
is an algebra homomorphism.
skow
we shallvthe kernel
one-to-one:
&t
of
rp
is the zero
n
cp
be in the kernel of
therefore,
= AI;
A =
-v
ak~·
~J q q p p
a ..
q=J..
n
n
Novl
;
n
n
Therefore,
ijql"'q I"'k
n
~~ 1
=
q=l
>'-'
A. K
N
n
n
)
)
c{J=
(AI)
n
1J
a.a . f3
~
r~s
=
rs
rand s,
i~l
,'OJ
a.
~
~I
••
~
n
0[3
a.a .
~
A =
•
n
s=l r=l
Therefore, for all
A
Then
r~s
= o.
rs
•
Since
11
has an identity
I, there exist combinations of the subscripts rand s so that for all i,
a.
r~s
#
O.
Therefore
(f) is an algebra monomorphism.
y
11 has no unity
Case 2.
element.
Imbed ~ into a ring ~*
with unity
and use the representation from Case 1.
Definition
Hi
that
N
35.
= Rl
,
Let
R
1
and HZ
be rings.
If R2 has a subring
R*
1
so
Rl is said to be imbedded in RZ"
then
There is a well knot-ill theorem that every ring can be imbedded into
a ring with identity.
1
1
Jacobson, Nathan, Lectures in Abstract Algebra, Vol. I, rage
86.
21
Example:
2 x 2
A faithful representation of the complex numbers as
real matrices.
The complex numbers are of dimension 2 over the real
numbers (we may think of complex numbers as ordered pairs
a and b are real numbers; and
=
(a, b) • (c, d)
(a,b)
(a,b) + (c,d) ,.. (a + c, b + d)
where
and
(ac - bd, ad + be).)
There is a natural basis for complex numbers over R1 :
66 =
table:
•
(1,0)
(0,1)
{l
= (1,0)
(1,0)
(0,1)
(1,0)
f "(Xlll (X12l
(0,1)
and
a'"
1 -.
i ..
bl o b
(-1,0)
.JC '\
(X2l2 (X222
'\
= b2
}
with multiplication
b ' b = (Xlllb + (X12l b 2 = 1(1,0)
l l
l
+ 0(0,1) ,.. (1,0)
t "-
(0,1)
~
i,.. (0,1)
(X112 (X122
(X21l (X221
Then,
• bl ,
2
= (X112 b1
+ (X122 b 2
= (0,1)
+ 1(0,1)
b 2 ·bl
= (X211 b1
+ (X221 b 2
+ 1(0,1)
(~ ~)
(-~ ~)
b 2 -b
2
= (X2l2 b l
Then
2
= 0(1,0)
= (0,1)
+ (X222 b 2
+ 0(0,1)
Now we consider the complex cube roots of 1.
-1 + {':3
and b'" 'W" 2 = -1 LJ =
2
= 0(1,0)
= -1{1,0)
= (-1,0).
J73
(that is, a is represented by the matriX)
J3/2)
-1/2
a =(
-./3/2
and by matrix multiplication,
-1/2
./3/2)
-1/2
-./3/2)
-1/2
(
-J3/2
J3/2)
,.. (1/4 - 3/4
-1/2
= b.
-1/2
Similarly
b
2
= a,
and of course
a3
= 1 = b3•
3/2
-3/2
-3/4 +
22
This gives us the table below which is clearly a finite cyclic group of
order 3.
• e
a
b
e
e
a
b
a
a
b
e
b
b
e
a
v.
Conclusion
To conclude this paper, I shall emphasize a point made in the introduction, namely that all of this abstract theory is not merely a mathematical game or fantasy.
In the field of quantum mechanics, the permu-
tation group helps explain the structure of the periodic table, energy
and momentum laws of quantum physics, and wave theory.
The symmetric
permutation group is a vital tool in the perturbation theory for the
construction of molecules, spin, and valence.
theoretic classification of atomic spectra.
There also is a group
Although one of the oldest
fields of mathematics, group theory has far reaching applications in
modern day science.
23
Bibliography
Barnes, Wilfred, Introduction to Abstract Algebra, Boston:
Heath and Co., 1963.--
D. C.
Burnside, William, Theory 2i Groups 2! Finite Order, New York:
Dover Publicatio~ Inc., 1955.
Burrow, Martin, Representation Theory
Academic Press, 1965.
21
Finite Groups, New York:
Curtis, Charles W. and Reiner, Irving, Representation Theory of Finite
Groups ~ Associative Algebras, New York: Interscience
Publishers, 1962.
Deskins, W. E., Abstract Algebra, New York:
The Macmillan Company, 1966.
Dickson, Leonard Eugene, Algebras ~ their Arithmetics, New York:
Dover Publications, Inc., 1960.
Hall, Marshall, 1h! Theory £i Groups, New Yorks
Company, 1959.
Herstein, I. N., Topics
1964.
~
Algebra, New York:
The Macmillan
Blaisdell Publishing Co.,
Jacobson, Nathan, Lectures 1a Abstract Algebra, Vol. I, New York:
D. Van Nostrand Company, Inc., 1951.
Scott, W. R.,Group Theory, Englewood Cliffs, New Jersey, Prentice-Hall,
Inc., 1964.
Wedderburn, J. H., Lectures on Matrices, New York:
Inc., 1964.
--
Dover Publications,
Wielandt, Helmut, translated by R. Bercov, Finite Permutation Groups,
New York: Academic Press, 1964.