Abstract Algebra
muM
Well
orderi
Every
ng principle
set of natural
con-empty
*
numbers
# E0 1 2 3,
, , ,
...
n3 has a least element .
Proof .
(Induction on the size of the set
Base case Let S5N with 1Sl = / a =
bgor ,
and Orcb .
Then S :
En3 for some neW
Thus
,
n is the least element in Proof
.
Let S =
Ga-bk/ke #, n - bk 03
S .
Suppose 8 S Then . a -
bk = 0 = a= bk
Ind Case Let Ke I" suppose every non-empty we can choose ro and t
-
.
,
subset of N of size K has
,
a least element Suppose OxS. Note that a so
,
and
if aco ,
a - b(2a) =
a(l 2b) >, -
Now let ,
S'sN with 15 : 101 thus a-blaaJeS ,
and S is
non-empty.
SinceK20132 there is an Since SSN and S is
non-empty,
,
,
element acS' and S" := S'15a3, S has a least element ,
res
,
r = a-bg,
which subset of N. for I
ge .
is a non-empty some
size K . By the induction hypothesis,
S"has a least element b
.
,
Clearly ,
r =0 and a =
bgbr Suppose .
BWOC,
that r = b .
Thus ,
minda b3 ,
is the least Then b(q01)
a
bq
- =
a -
- b =
r -
b = 0
element of S TJ and a-blgp1) by < a - = r *
contradiction.
r = min(S) ,
so there cannot be an
element of S that is less them r.
So rub .
7 such that
Sps 79 ,, 92 ,
r ,
r
,
a =
ba tr , ,
a =
bactre ,
and Dr
,
r ,b
WLOG that Then,
, suppose r
,
3r
..
If alb
,
a is a divisor of b bq ,
br
,
=
baybre
=
b(q,92) =
my r
,
and b is a
multiple of a So ,
blire-r ) , .
However ,
Ostr ,
I re
=> r
=r,
= 0 = r = r * unique .
,
So ,
a -baz =
a -
by ,
92 =
9 ,
* unique
,Definition .
A
symmetry of a 2-D shope/ Definition Va b , ,
ceDn ,
(ab)c albe) =
,
Dn is
operation that associative.
region is an on
object s .
6 .
the imap befores
after the operation or identical.
A B
Symmetrics of square a Definition ↑
Let X G he sets. A
binay
>
-
Ro(rotation 5 coul CD
operation on
G is a function,
Ras (rotation 939
Ris (rotation 1800 · : > X
GxG -
Reo (rotation 270)
%
It (Glip about horizontal) Note Instead of (a b) ,
we write ab
verticle)
,
v
(flip about or .
ab
D
AD Iflip about disul AD)
BC
-
Disc (flip about
dapul If XSG
,
then we say G is closed
under
Every other symmetry is
equivalent to one of
Example Let In := E(0] [1, In-13. The , ...
these .
8 addition mode is a binay
operation on In .
Also closed ,
Definition The dihedral of order
group zn
.
(D.) the
LetGbe asetwihdin. )
of symmetries Definition
is
group
of a
regula n-gon . . ,
GroupIf
the
is a the
following
Definition ↑
The Cayley Table is complete axioms hold.
multiplication table of all elements Closin .
0 G is closed under
in the
group .
Associativity 1 . Va b cEG , , ,
(ab)c = a(bc)
Identity 2 JecG
=
.
,
ac-ea = a
·
Inverse .
3 JacG ca = a .
a = 2
o ,
Abelian Group
08
·
O
Commutativity 4 .
a -
b = b .
a
0
0
O sometime the
binary operation is rotated as
*
column o row not commutative.
,
o inverses are commutative .
Since Dn is not commutative ,
Dn is
non-Abelian
, In
Thorm .
a
group
G
,
the identity is Order and
Subgroups
Definition The number of elements
unique. . in a
G is the order of G demoed
group
, , ,
pf .
Suppose e
,
e, G ze dentity
elements. Then
,
e
,
= 2
,
e
,
=
e
Example (( 011 .
,
= -
Theorem Let G be Then for all I(V , )) =
group
.
a .
a b ceG,
Un=Eme(god(m n)
, ,
ab =
ac = b = e (left cancellation) Recall . ,
= 13 is a
ba =a => b=c (right cancellation group
w/ multiplication .
ab-ac Let a G.
pf Suppose .
.
Thus ,
Up =
21 ,
2 4
, ,
5
,
7
,
83 /Val ,
= 9
b = eb
-(a a) Example 131 . ,
-1
,
:
, :3) = 4
= (ab)
a (ac)
=
Definition .
The order of an element
geG is
(a ale
"
the smallest positive
=
such
inter n
eC that 1 e
g
=
ng-o-e
= or
b =
c
,
Likemse with ba- ca T
In Vo 111 = 1
,
131 = 4 171 =
4
,
191 = 2
, ,
Theorem If G a ,beG then
group and
.
is a
,
(ab)" :3
,
= b a In E1 ,
-1
,
i
,-
111 - 1 ,
1-11 52
,
,
Let consider abeG Since (i) 4 1 - i) 4
pf a beG and = =
.
.
, , ,
G is closed under immuses and multipleation .
b a EG Definition. if G is a and acG,
group
Earlne * 3
,
Then (b a )(ab) =
b (a a)b b b =
then (a ) .
=
Enalne 13
,
b "(b) =
b (eb) = =
e (a) =
Similarly (ab)(b a-1) e =
T
Definition If a subset HSG is a
group
.
Theorem For .
each aEG ,
thre is a
unique elevant under the same operation as
beG that ab-ba (Inverses G It
,
such = e .
or
,
we say is a
subgroup
nuique) .
Denoted HSG .
pf .
Suppose b ,beeG ,
and ab =b , ,
a = e
Then ab abz = b Definition A proper of G
and abe = b,a = e .
= ,
= be , .
subgroup a
group
is
by left cancellation H
.
A a
subgroupIs G such thut * G
denoted H<G.
subgroup Ge3
For G the is
group ,
the trivial If HSG
subgroup of G .
and H*Ge3 ,
It is called non-trivial .
, Example
. 50 23 ,
<
Ey :
Proper subgroup Theorem 2-Step
.
subgroup test
:
>
-
Closed Let G be a and ** HSG.
group
Associative If It is closed under the operation of G,
Identity and has inverses ,
then HSG
Inverses p If It is closed under "multiplication" and
taking inveses then a, bel = abelt.
,
Thus by
, 1-step test ,
HSG. TJ
Suppose groupand
a
Theorem .
ExampleLearn
. Seder x
p . Notice associativity of in It is
inherited from (G.. ) .
Since H **,
let xeH .
Choosing a = b = X
, gives
as
XX" ab" H Next
e = =
.
,
choosing
Theorem. Finite test :
a = e and b = x
gives as
LetI be
Subgroup
X = e . X = abel for all XeHt . a nommpty ,
finite subset
of G T It is closed
group
.
Next let
,
x
, yelt. We know y "is under the operation of G.
I
in It ,
and a = x
,
b =
y gives
Xy = x(y)) = ab H pf .
Let x+ H. Firstf x+e ,
then clearly
X etH= .
Example Let G be an abelian and let Suppose X* C and consider the
-
group
H =
ExcG)x e3 =
.
Show HSG. elements X
,
X2 , X3 ,
...
eG .
Since It is
Associativity Inherited from
*
: G closed under multiplication ,
X H.
Identity e :
= e
,
H Since It is finike, Si , je &" , wher isj
Inverses (x-1)" (x2)" e
↑
: = = = e and Xi = X! .
Closure :
a belt b = e b b
. =
, , ,
lab)" = ab' *
Associativity is Commutativity Thus ,
xi =
Xi = xi i =
e .
= l l .
= C Since xxe
,
i -
j > /
i
xi i =
xi
-
=>
-
x . = = e
Note e'=e ,
and celt. So H .
* D
,
Suppose a be H .
Thus ate = b? This So ,
x = xi -T !. Thus XeH.
,
means a = a is
&
b" = b .
Since Gis Abelian
and associative :
(ab") (ab)" (ab)(ab) : = = (a a)(b b)
- - =
ab
=
e .
e =
c
. Thus HSG ·
$
muM
Well
orderi
Every
ng principle
set of natural
con-empty
*
numbers
# E0 1 2 3,
, , ,
...
n3 has a least element .
Proof .
(Induction on the size of the set
Base case Let S5N with 1Sl = / a =
bgor ,
and Orcb .
Then S :
En3 for some neW
Thus
,
n is the least element in Proof
.
Let S =
Ga-bk/ke #, n - bk 03
S .
Suppose 8 S Then . a -
bk = 0 = a= bk
Ind Case Let Ke I" suppose every non-empty we can choose ro and t
-
.
,
subset of N of size K has
,
a least element Suppose OxS. Note that a so
,
and
if aco ,
a - b(2a) =
a(l 2b) >, -
Now let ,
S'sN with 15 : 101 thus a-blaaJeS ,
and S is
non-empty.
SinceK20132 there is an Since SSN and S is
non-empty,
,
,
element acS' and S" := S'15a3, S has a least element ,
res
,
r = a-bg,
which subset of N. for I
ge .
is a non-empty some
size K . By the induction hypothesis,
S"has a least element b
.
,
Clearly ,
r =0 and a =
bgbr Suppose .
BWOC,
that r = b .
Thus ,
minda b3 ,
is the least Then b(q01)
a
bq
- =
a -
- b =
r -
b = 0
element of S TJ and a-blgp1) by < a - = r *
contradiction.
r = min(S) ,
so there cannot be an
element of S that is less them r.
So rub .
7 such that
Sps 79 ,, 92 ,
r ,
r
,
a =
ba tr , ,
a =
bactre ,
and Dr
,
r ,b
WLOG that Then,
, suppose r
,
3r
..
If alb
,
a is a divisor of b bq ,
br
,
=
baybre
=
b(q,92) =
my r
,
and b is a
multiple of a So ,
blire-r ) , .
However ,
Ostr ,
I re
=> r
=r,
= 0 = r = r * unique .
,
So ,
a -baz =
a -
by ,
92 =
9 ,
* unique
,Definition .
A
symmetry of a 2-D shope/ Definition Va b , ,
ceDn ,
(ab)c albe) =
,
Dn is
operation that associative.
region is an on
object s .
6 .
the imap befores
after the operation or identical.
A B
Symmetrics of square a Definition ↑
Let X G he sets. A
binay
>
-
Ro(rotation 5 coul CD
operation on
G is a function,
Ras (rotation 939
Ris (rotation 1800 · : > X
GxG -
Reo (rotation 270)
%
It (Glip about horizontal) Note Instead of (a b) ,
we write ab
verticle)
,
v
(flip about or .
ab
D
AD Iflip about disul AD)
BC
-
Disc (flip about
dapul If XSG
,
then we say G is closed
under
Every other symmetry is
equivalent to one of
Example Let In := E(0] [1, In-13. The , ...
these .
8 addition mode is a binay
operation on In .
Also closed ,
Definition The dihedral of order
group zn
.
(D.) the
LetGbe asetwihdin. )
of symmetries Definition
is
group
of a
regula n-gon . . ,
GroupIf
the
is a the
following
Definition ↑
The Cayley Table is complete axioms hold.
multiplication table of all elements Closin .
0 G is closed under
in the
group .
Associativity 1 . Va b cEG , , ,
(ab)c = a(bc)
Identity 2 JecG
=
.
,
ac-ea = a
·
Inverse .
3 JacG ca = a .
a = 2
o ,
Abelian Group
08
·
O
Commutativity 4 .
a -
b = b .
a
0
0
O sometime the
binary operation is rotated as
*
column o row not commutative.
,
o inverses are commutative .
Since Dn is not commutative ,
Dn is
non-Abelian
, In
Thorm .
a
group
G
,
the identity is Order and
Subgroups
Definition The number of elements
unique. . in a
G is the order of G demoed
group
, , ,
pf .
Suppose e
,
e, G ze dentity
elements. Then
,
e
,
= 2
,
e
,
=
e
Example (( 011 .
,
= -
Theorem Let G be Then for all I(V , )) =
group
.
a .
a b ceG,
Un=Eme(god(m n)
, ,
ab =
ac = b = e (left cancellation) Recall . ,
= 13 is a
ba =a => b=c (right cancellation group
w/ multiplication .
ab-ac Let a G.
pf Suppose .
.
Thus ,
Up =
21 ,
2 4
, ,
5
,
7
,
83 /Val ,
= 9
b = eb
-(a a) Example 131 . ,
-1
,
:
, :3) = 4
= (ab)
a (ac)
=
Definition .
The order of an element
geG is
(a ale
"
the smallest positive
=
such
inter n
eC that 1 e
g
=
ng-o-e
= or
b =
c
,
Likemse with ba- ca T
In Vo 111 = 1
,
131 = 4 171 =
4
,
191 = 2
, ,
Theorem If G a ,beG then
group and
.
is a
,
(ab)" :3
,
= b a In E1 ,
-1
,
i
,-
111 - 1 ,
1-11 52
,
,
Let consider abeG Since (i) 4 1 - i) 4
pf a beG and = =
.
.
, , ,
G is closed under immuses and multipleation .
b a EG Definition. if G is a and acG,
group
Earlne * 3
,
Then (b a )(ab) =
b (a a)b b b =
then (a ) .
=
Enalne 13
,
b "(b) =
b (eb) = =
e (a) =
Similarly (ab)(b a-1) e =
T
Definition If a subset HSG is a
group
.
Theorem For .
each aEG ,
thre is a
unique elevant under the same operation as
beG that ab-ba (Inverses G It
,
such = e .
or
,
we say is a
subgroup
nuique) .
Denoted HSG .
pf .
Suppose b ,beeG ,
and ab =b , ,
a = e
Then ab abz = b Definition A proper of G
and abe = b,a = e .
= ,
= be , .
subgroup a
group
is
by left cancellation H
.
A a
subgroupIs G such thut * G
denoted H<G.
subgroup Ge3
For G the is
group ,
the trivial If HSG
subgroup of G .
and H*Ge3 ,
It is called non-trivial .
, Example
. 50 23 ,
<
Ey :
Proper subgroup Theorem 2-Step
.
subgroup test
:
>
-
Closed Let G be a and ** HSG.
group
Associative If It is closed under the operation of G,
Identity and has inverses ,
then HSG
Inverses p If It is closed under "multiplication" and
taking inveses then a, bel = abelt.
,
Thus by
, 1-step test ,
HSG. TJ
Suppose groupand
a
Theorem .
ExampleLearn
. Seder x
p . Notice associativity of in It is
inherited from (G.. ) .
Since H **,
let xeH .
Choosing a = b = X
, gives
as
XX" ab" H Next
e = =
.
,
choosing
Theorem. Finite test :
a = e and b = x
gives as
LetI be
Subgroup
X = e . X = abel for all XeHt . a nommpty ,
finite subset
of G T It is closed
group
.
Next let
,
x
, yelt. We know y "is under the operation of G.
I
in It ,
and a = x
,
b =
y gives
Xy = x(y)) = ab H pf .
Let x+ H. Firstf x+e ,
then clearly
X etH= .
Example Let G be an abelian and let Suppose X* C and consider the
-
group
H =
ExcG)x e3 =
.
Show HSG. elements X
,
X2 , X3 ,
...
eG .
Since It is
Associativity Inherited from
*
: G closed under multiplication ,
X H.
Identity e :
= e
,
H Since It is finike, Si , je &" , wher isj
Inverses (x-1)" (x2)" e
↑
: = = = e and Xi = X! .
Closure :
a belt b = e b b
. =
, , ,
lab)" = ab' *
Associativity is Commutativity Thus ,
xi =
Xi = xi i =
e .
= l l .
= C Since xxe
,
i -
j > /
i
xi i =
xi
-
=>
-
x . = = e
Note e'=e ,
and celt. So H .
* D
,
Suppose a be H .
Thus ate = b? This So ,
x = xi -T !. Thus XeH.
,
means a = a is
&
b" = b .
Since Gis Abelian
and associative :
(ab") (ab)" (ab)(ab) : = = (a a)(b b)
- - =
ab
=
e .
e =
c
. Thus HSG ·
$
