Lecture Syllabus in Abstract Algebra

8/11/2019 Lecture Syllabus in Abstract Algebra

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REVIEW

Definition. Let A and B be non-empty sets. Arelation from A to B is a set of ordered pairs ( a,b ) where Aa and Bb .

Definition. A relation R on a set A is an equivalencerelation in A provided:i. R is reflexive, that is, R x x , for all x

in A.ii. R is symmetric, that is, R y x ,

implies R x y , .iii. R is transitive, that is, R y x , and

R z y , imply R z x , .

Example. Let Z be the set of integers and m afixed positive integer. The relation ofcongruence modulo m in Z is an

equivalence relation in Z.


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Definition. Let A and B be non-empty sets. Afunction f from A to B is a relation from

A to B such that:

i. The domain of f is A.ii. If f y x , and f z x , then y=z.

The image of f , denoted by im f or f (A),is the set Aaa f )( .

Definition. The function f is onto or surjective ifim f = B, that is, for each Bb thereexists Aa such that b = f (a).

Definition. The function f is one-to-one or injective provided that, for all A y x , such that

y x , then )()( y f x f .

Definition. The function f is bijective if it is bothinjective and surjective.


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BINARY OPERATIONS AND STRUCTURES

Definition. Let S be a non-empty set. A binaryoperation on S is a function from S x Sinto S.

Definition. A non-empty set together with a binaryoperation * defined on it is called agroupoid, denoted by (G,*).

Definition. A binary operation * on a non-empty setG is said to be commutative provideda*b = b*a for all a,b in G.

Definition. A binary operation * on a non-empty setG is said to be associative provideda*(b*c) = (a*b)*c for all a,b,c in G.

Definition. A semigroup is a groupoid (G,*) inwhich * is associative.


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Definition. A monoid is a semigroup G whichcontains an element e such that ae = ea

= a for all a in G. The element e is calledthe two-sided identity element of G.

Definition. A group is a monoid G such that, forevery a G, there exists x G such thatax = xa = e. The element x is called thetwo-sided inverse of a and is usuallydenoted by a -1.

Definition. A group G with a commutative binaryoperation is called an abelian group.

Examples of groups:1. The set of integers Z under addition2. The set of rational numbers Q under addition3. The set of real numbers R under addition

4. The set of complex numbers C under addition5. The set U = {z C | z 4 1 = 0} under complex

multiplication


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6. The set Z m = {0,1,2,3,...,m-1} under additionmodulo m, where m Z +

7.

The set of nxn matrices over R under matrixaddition, where n Z+ 8. The cross product Z 2 x Z 2 under component-

wise addition modulo 29. The set of symmetries of an equilateral triangle10. The set of symmetries of a square

Examples of monoids which are not groups:1. The set of integers Z under multiplication2. The set of rational numbers Q under

multiplication3. The set of real numbers R under multiplication4. The set Z m = {0,1,2,3,...,m-1} under

multiplication modulo m, where m Z+ 5. The set of complex numbers C under

multiplication

6. The set of nxn matrices over R under matrixmultiplication where n Z+

7. The cross product Z 2 x Z 2 under component-wisemultiplication modulo 2


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Example of a semigroup which is not a monoid:

1.

the set of even integers under multiplication

SEMIGROUPS, MONOIDS AND GROUPS

Theorem. If G is a monoid, then the identityelement is unique.

Theorem. Let G be a group. Then:i. and imply

ii. The inverse of an element is unique.iii. Left and right cancellation laws hold

in G.iv. For each ( ) v. For ( )

vi. For each the equations and have unique

solutions.


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Theorem. Let G be a semigroup. Then G is agroup if and only if for all the

following conditions hold:i. There exists an element suchthat for all

ii. For each , there exists anelement such that

SUBGROUPS

Definition. Let G be a group and H a non-emptysubset that is closed under the product inG. If H is itself a group under the

product in G, then H is said to be asubgroup of G. This is denoted by

Theorem. Let H be a non-empty subset of a group

G. Then if and only if for all .


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COSETS AND NORMALS

Definition. Let H be a subgroup of a group G, and Then a is right congruent to b modulo H , denoted if

Further, a is left congruent to b modulo H, denoted if

Remarks:1. If G is abelian, then left and right congruence

modulo H coincide.2. There exist non-abelian groups G and subgroups

H such that left and right congruence modulo H coincide, but this is not true in general.

3. Left (resp. right) congruence modulo H is anequivalence relation on G.

4. For right congruence modulo H , the equivalence

class of is the set { } andis called a right coset of H containing a.


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5. For left congruence modulo H , the equivalenceclass of is the set { } andis called a left coset of H containing a.6. A left coset is not necessarily a right coset. Ingeneral,

7. It may happen that even if

Theorem. Let 1. 2. if and only if 3. if and only if 4. Either or 5.

6. Let L be the set of distinct left cosets

of H in G, and R the set of distinctright cosets of H in G. Then | L|=| R|.

7.

Definition. Let The index of H in G,denoted [ ] is the number distinctleft (resp. right) cosets of H in G.


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Theorem. (Lagrange) Let . Then

[ ] In particular, if G is finite,then divides and further, theorder | a | of divides | G|.

Note: The converse of this theorem is notnecessarily true.

Definition. A subgroup H of a group G is said to benormal if the relations of leftcongruence modulo H and rightcongruence modulo H coincide.

Remarks: If N is normal in G, then:1. Every left coset of N in G is a right coset of N

in G, and conversely.2. 3.

Theorem. Let . Then N is normal in G if andonly if


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Theorem. Every subgroup of an abelian group isnormal.

Theorem. Let If [ ] then H isnormal in G.

Remark. Normality is not transitive. However itis easy to see that if N is normal in G,then N is normal in every subgroup of G that contains N .

QUOTIENTS

Theorem. Let R(

) be an equivalence relation on amonoid G with the property that

a b and cd imply ac bd.

Then the quotient class is a monoid

under the binary operation [a][b]=[ab],where [a] denotes the equivalence class


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of . If G is an [abelian] group,

then so is .

Definition. An equivalence relation that satisfies theadditional condition in the abovetheorem is called a congruence relation.

Examples:

1. The quotient classm

Z = {[0], [1], [2],.....,[m-1]}

under the operation [x] + [y] = [x+y] is a finiteadditive abelian group with m elements.

2. The quotient classm

Z = {[0], [1], [2], ...,[m-1]}

under the operation [x] [y] = [xy] is a finitemultiplicative and commutative monoid with m elements which is not a group in general.

Theorem. Let N be a normal subgroup of G. Then

right (resp. left) congruence modulo N isa congruence relation on G.


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Theorem. If N is normal in G and is the set of

all (left) cosets of N in G, then is agroup of order [ ] under the binaryoperation ( )( )

DIRECT PRODUCT AND SUM

Definition. Let G and H be groups. The direct product of G and H, denoted by GxH , isthe group whose underlying set is GxH and whose binary operation is given by

(a,b)(c,d)=(ac,bd) where a,c G and b,d H .

Remarks.1. GxH is abelian if and only if G and H are abelian.2. |GxH| = |G| |H|3. We use the additive operation G H if the

operation in G and H are commutative, and call itdirect sum of G and H.


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4. The direct product can be extended to more thantwo groups.

RINGS, INTEGRAL DOMAINS AND FIELDS

Definition. A ring is a nonempty set R together withtwo binary operations (usually denotedas addition (+) and multiplication) suchthat:i. (R,+) is an abelian group;

ii. Multiplication is associative;iii. a(b+c) = ab+ac and (a+b)c = ac + bc

for all a,b,c in R.

Note: The pair (R,+) is called the additivegroup of R.

Definition. A ring R is said to be a commutative

ring if multiplication in R iscommutative.


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Note: The additive identity of the ring R iscalled zero element or zero, and is

denoted by 0 R .

Definition. A ring R is said to be a ring with unityor a ring with identity if (R,) is amonoid. The multiplicative identity, orsimply identity or unity, is denoted by1R .

Definition. An element of a ring R with unity with atwo-sided multiplicative inverse iscalled a unit.

Remark. The left and right inverses of a unit in aring with identity coincide.

Theorem. The set of units in a ring with identity

forms a multiplicative group.

Definition. A ring R with identity is said to be a division ring if (R-{0},) is a group.


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Remark. Every division ring has at least two

elements (namely 0 and 1).

Definition. A field is a commutative division ring.

Example. The ring Z p, p prime, under addition andmultiplication mod p, is a field.

Definition. A nonzero element a in a ring R is saidto be a left [resp. right] zero divisor ifthere exists a nonzero such thatab = 0 [resp. ba = 0]. A zero divisor isan element of R which is both a left anda right zero divisor.

Theorem. A ring R has no zero divisors if and onlyleft and right (multiplicative)

cancellation laws hold in R.


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Definition. A commutative ring R with unity and no zero divisors is called an integral

domain.

Remark. A unit is not a zero divisor, andconversely.

Example. The ring (Z,+, ) is an integral domain.

Example. The ring Z p, p prime, under addition andmultiplication mod p, is an integraldomain.

Remarks. Every integral domain has at least twoelements (namely 0 and 1).

.Theorem. A field is an integral domain.Conversely, a finite integral domain is a

field.

Examples. The four different quaternarycommutative rings with unity.


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MODULES AND VECTOR SPACES

Definition. Let R be a ring. A (left) R-module is anadditive abelian group A together with afunction A A R (the image of (r,a)

being denoted by ra) such that for allr,s R and a,b A:

i. r(a+b) = ra+rbii. (r+s)a = ra+sa

iii. r(sa) = (rs)a

If R has an identity 1 R such that 1 R a = a

for all a A, then A is said to be aunitary (left) R-module.

If R is a division ring, then a unitary(left) R-module is called a (left) vector

space.


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Definition. A subfield K of a field F is a subring of

F that is a field itself. In such a case, wecall F an extension field of K (or simplyan extension of K ).

Definition. Let R be a ring, A an R-module and B anon-empty subset of A. B is said to be asubmodule of A provided that:

i. B is an additive subgroup of A;ii. for all .

A submodule of a vector space over adivision ring is called a subspace.

Theorem. Let B be a non-empty subset of amodule A over a ring R. Then B is an R-

submodule of A if and only if and only if for all and


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ORDER AND CHARACTERISTIC

Definition. The order of a group G is the cardinalnumber | G|. G is said to be finite[resp.infinite] if | G| is finite[resp.infinite].

Definition. Let G be a group and . The orderof a, denoted by is the least positiveinteger m such that If no such

positive integer m exists, we say that a isof infinite order.

Definition. An element of a group of order 2 iscalled an involution. That is, aninvolution is a non-identity elementwhose inverse is itself.

Remark: The identity of the group is the onlyelement of order 1.


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Theorem. Let a be an element of a group of finiteorder. Then

Definition. Let R be a ring. If there is a least positive integer n such that forall , then R is said to havecharacteristic n. If no such n exists, then

R is said to have characteristic 0.

Theorem. Let R be a ring with unity . If n is theleast positive integer such that then char( R) = n.

Remark: The characteristic of R is the order of in the additive group of R.

HOMOMORPHISMS

Definition. Let ( ) and ( ) be semigroups. Afunction is a homomorphism provided that ( ) ( ) ( ) for all


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ii. iii. ( ) ( ) for all iv.

is a monomorphism if and only ifker f = { }

If f is an isomorphism, then:v.

vi. G is abelian if and only if H isabelian.

vii. ( ) for all

Definition. Let R and S be rings. A function

is a homomorphism of rings provided that for all i. ( ) ( ) ( )

ii. ( ) ( ) ( ) Remarks: A homomorphism of rings is, in

particular, a homomorphism of theunderlying additive groups.


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A monomorphim of rings is also calledan embedding of R in S .

The kernel of a ring homomorphism fis its kernel as a map of additive groups,that is, ker f = { ( ) }

Definition. Let A and B be modules over a ring R.A function is an R-modulehomomorphism provided that for all

and i. ( ) ( ) ( )

ii. ( ) ( )

If R is a division ring, then an R-modulehomomorphism is called a lineartransformation.

The kernel of an R-module homomorphism f is its kernel as a map ofadditive groups, that is,ker f = { ( ) }.


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Theorem. If is a homomorphism ofgroups, then is normal in G.Conversely, if N is normal in G, then the

map given by ( ) is an epimorphism with kernel N .

CYCLIC GROUPS AND PRINCIPAL IDEALS

Definition. A group G is said to be cyclic if thereexists such that { } In this case, we denote G by andthe element a is called a generator of G.

Remarks:1. Every cyclic group is abelian.

2. A non-cyclic group may have a cyclicsubgroup.3. A subgroup of a cyclic group is cyclic.


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4. The order of an element of a group G is theorder of the cyclic subgroup

Examples.1. The Klein-4 group is not cyclic.2. The quaternion group is not cyclic but all its

proper subgroups are cyclic.

Theorem. Let be a cyclic group. If G isinfinite, then a and are the onlygenerators of G. If G is finite of order n, then if and only if ( )

Theorem. Every infinite cyclic group isisomorphic to the additive group Z andevery finite cyclic group of order m isisomorphic to the additive group

Definition. A subring I of a ring R is a left ideal provided and imply A subring I of a ring R is a right ideal

provided and imply


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I is an ideal if it is both a left ideal and aright ideal.

Definition. A [left] ideal I of R such that and is called a proper [left] ideal.

Theorem. A nonempty subset I of a ring R is a left[resp. right] ideal if and only if for all

and

i. implies andii. and imply [resp.

]

Theorem. If R has an identity and I is a [left]ideal of R, then if and only if

Consequently, a nonzero [left]ideal I of R is proper if and only if I

contains no units of R.


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Theorem. Let R be a ring and I an ideal of R. Then

the additive quotient group is a ring

with multiplication given by( )( ) .

If R is commutative or has an identity,

then the same is true for .

Theorem. If is a ring homomorphism,then the kernel of f is an ideal of R.Conversely, if I is an ideal in R, then the

map given by ( ) is an epimorphism of rings with kernel I .

Definition. Let R be a commutative ring with unity.The principal ideal ( ) is the set{ } This ideal is also denoted by

Ra or aR. If the ideals of R are all principal, then R is called a principalideal ring.


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Examples. and are principal ideal rings.

CENTER OF A GROUP OR RING

Definition. Let ( ) be a group. The center of G,denoted by Z(G), is the set ( ){ }.

Remark. Z(G) is an abelian subgroup of G.Further, G is abelian if and only if Z(G)= G.

Definition. Let ( ) be a ring. The center of R,denoted by ( ) , is the set ( ){ }.

Remark. ( ) is improper since R is an abeliangroup. ( ) is a subring of R, but maynot be an ideal. Further, R iscommutative if and only if ( )


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PRIME AND MAXIMAL IDEALS

Definition. Let P be an ideal of a commutative ring R such that . Then P is said to be prime if and only if, for all

or

Remarks.1. The zero ideal in any integral domain is prime.2. The prime ideals of Z are {0} and ( p), where p is

prime.

Theorem. In a commutative ring R with anideal P is prime if and only if the

quotient ring is an integral domain.

Definition. An ideal [resp. left ideal] M in a ring R is said to be maximal if and forevery ideal [resp. left ideal] N such that

, either or .


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Theorem. Let M be an ideal in a commutative ring R with Then M is maximal if

and only if is a field.

Theorem. The following conditions on acommutative ring R with areequivalent.

i. R is a field.ii. R has no proper ideals.

iii. {0} is a maximal ideal of R.

ISOMORPHISM THEOREMS

Theorem: If is a homomorphism ofgroups and N is a normal subgroup of G contained in the kernel of f , then there isa unique homomorphism

such that ( ) ( ) for all and ( )


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is an isomorphism if and only if f is anepimorphism and

Corollary: If is a homomorphism ofgroups, then f induces an isomorphism

.

Corollary: If K and N are subgroups of a group G,

with N normal in G, then ( )

Corollary: If H and K are normal subgroups of agroup G such that then is a

normal subgroup of and( ) ( )

Theorem: If is a homomorphism of ringsand I is an ideal of R which is containedin the kernel of f , then there is a uniquehomomorphism such that ( ) ( ) for all

and ( ) is an


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isomorphism if and only if f is anepimorphism and

Corollary: If is a homomorphism of rings,then f induces an isomorphism of rings

.

Corollary: There is an isomorphism of rings

Corollary: If is contained in J , then is an idealin and ( ) ( )

PERMUTATION GROUPS

Definition. Let X be a nonempty set. A permutationof X is a bijection from X onto itself.

Theorem. Let Sym(X) be the set of all permutationson X. Then Sym(X) is a group undercomposition.


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Definition. If X = {1,2,3,, n}, then Sym(X) iscalled the symmetric group of n letters,

denoted by S n.

Remark:

Theorem. Every permutation in S n can be writtenas a product of (not necessarily disjoint)transpositions.

Definition. A permutation is even (resp. odd) if itcan be written as a product of an even(resp. odd) number of transpositions.

Theorem. The set An of all even permutations is anormal subgroup of S n of index 2.Furthermore, An is the only subgroup ofS n of index 2.

Remark. An is called the alternating group ofdegree n.


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Definition. The dihedral group Dn of degree n is asubgroup of S n (n 3) generated by

( ) and

Remark: Dn has 2n elements and is usuallyassociated with the group of all

symmetries on a regular polygon with nsides.

Lecture Syllabus in Abstract Algebra

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