Abstract Algebra

Complex numbers - [ ] Polar representation of complex numbers - [ ] The nth root of unity - [ ] Some simple geometric notions and properties - [ ] Conditions in collinearity, orthogonality, concyclicity, - [ ] Analytic geometry in the complex plane

Polynomials - [ ] Rational and integral polynomials - [ ] Factor theorem - [ ] Roots of a polynomial - [ ] Relation between roots and coefficients of a polynomial - [ ] Sum of powers of roots - [ ] Symmetric functions - [ ] Transformation of equations - [ ] Roots with signs changed - [ ] Synthetic division - [ ] Cardano’s solution of a cubic polynomial - [ ] Descartes’ solution of a biquadratic equation

Group Theory

Group

non empty set G with binary operation * is a group (G,*) if \forall a,b,c\in G, 1. [Groupoid] a*b \in G :Closed 2. [Semi Group] a*(b*c)=(a*b)*c :Associative 3. [Monoid Semi Group] \exists e\in G: a*e=e*a=a :Identity 4. [Group] \exists a'\in G: a*a'=a'*a=e :Inverse 5. [Abelian Group] a*b=b*a :Commutative

Order of a Group

O(G)= |G|= number of elements in (G,*)

Cayley Table or Composition Table

e.g. cube roots of unity, \omega^3=1; 1+\omega+\omega^2=0

\times	1	\omega	\omega^2
1	1	\omega	\omega^2
\omega	\omega	\omega^2	1
\omega^2	\omega^2	1	\omega

same row => identity = e = 1
if symmetric matrix => abelian group

Addition and Multiplication modulo m

(+_m, \times_m)

Cyclic Group

G=[a], \exists a\in G: every element in G can be expressed as some integral power (in case of multiplication operation) of a called the generator. - Generators for (G,+_m)=\{a: gcd(a, m)=1,\forall a\in G\} - If |G|=p^n, number of generators =p^n-p^{n-1}

Theorems on Cyclic Groups

Every cyclic group is abelian. (converse not necessarily true)
If a is generator of G, then a^{-1} is also a generator of G.
O(finite cyclic group) = O(generator of the group)
- Cor. A finite group with O(G)=n is cyclic if and only if it has an element of order n. (converse is true)
Every infinite cyclic group has two and only two generators.
Every subgroup of a cyclic group is also cyclic.
- Cor. Every proper subgroup of an infinite cyclic group is infinite.

Cosets

Let (H,*) \leq (G,*), then \forall a\in G the set aH=\{ah:h\in H\} is a left coset of H in G and Ha=\{ha:h\in H\} is a right coset of H in G.

aH,Ha \subset G\forall a\in G

Also, eH=H=He, i.e. the left and right cosets of H corresponding to the identity e coincide with H. Hence, H itself is a left as well as right coset of H in G.

there are n cosets of n\mathbb{Z} in (\mathbb{Z},+)

Theorems on cosets

If H\leq G and a\in G, then a\in aH and a\in Ha.
Any two left (or right) cosets of a subgroup are either identical or disjoint.
Lagrange’s Theorem The order of every subgroup of a finite group is a divisor of the order of the group. (converse not necessarily true)
- Cor. 1 The order of every element of a finite group is a divisor of the order of the group.
- Cor. 2 If G is a finite group of order n and a\in G, then a^n=e.
- Cor. 3 Every group of prime order is cyclic.

Normal (special, invariant, self conjugate) Subgroup

Let H\leq G, then H \lhd G \iff xHx^{-1} \subseteq H,\forall x\in G

Proper and Improper Normal Subgroup

For every group G we have at least two normal subgroups, 1. G itself 2. \{e\}

These two are called the improper normal subgroups. Any other normal subgroup is called a proper (\unlhd) normal subgroup.

Simple Group

Group whichi does not have a proper normal subgroup, e.g. every group of prime order

Hamiltonian Group

If all subgroups of the non-abelian group are normal

Theorems on normal subgroups

Every subgroup of an abelian group is a normal subgroup.
- Cor. Every subgroup of a cyclic group is a normal subgroup.
A subgroup H of a group G is a normal subgroup if and only if xHx^{-1}=H,\forall x\in G, i.e. H \lhd G \iff xHx^{-1} = H,\forall x\in G (converse is true)
A subgroup H of G is normal subgroup if and only if each left coset is equal to the right coset of H and vice versa. i.e. H \lhd G \iff xH=Hx,\forall x\in G (converse is true)
A subgroup H of a group G is a normal subgroup if and only if the product of the two right (or left) cosets of H in G is again a right (or left) coset of H in G. i.e. H \lhd G \iff HaHb=Hab, \forall a,b\in G (converse is true)

Quotient (Factor) Group

Let H\lhd G, then set G/H of all cosets of H in G together with the binary composition defined by HaHb=Hab where Ha,Hb\in G/H, is called a quotient group of G by H.

Theorems on quotient groups

Every quotient group of an abelian group is abelian but converse is not true. e.g. of converse not being true, (from permutation groups):
- S_3/A_3 has order 2
- every group of order 2 is abelian
- but S_3 is not abelian
Every quotient group of a cyclic group is cyclic group but not conversely.

![[common groups to remember.png]]

Homomorphism

A mapping f from a group (G,*) to (G',\cdot) is called a group homomorphism or a group morphism from G to G' if f(a*b)=f(a)\cdot f(b), \forall a,b\in G. Thus, if f is morphism from G to G', then it preserves the composition in both the groups i.e. image of the composite = composite of images.

Type of morphism	Condition
Monomorphism	f is injection
Epimorphism	f is surjection
Isomorphism	f is bijection
Endomorphism	G'=G i.e. f: G\to G
Automorphism	G'=G and f is bijection

Theorems on homomorphisms

If f is a homomorphism from G to G' and e and e' be their respective identities, then
1. f(e)=e'
2. f(a^{-1}=[f(a)]^{-1}, \forall a\in G
If f is a homomorphism from G to G', then
1. H\leq G\implies f(H) \leq G'
2. H'\leq G\implies f^{-1}(H')=\{x\in G:f(x)\in H'\}\leq G
- Cor. If f:G\to G' is a homomorphism, then f(G)\leq G'

Kernel of Homomorphism

Let f:(G,*)\to(G',\cdot), then Ker f= K = \{x\in G:f(x)=e'\in G'\} where e' is the identity of G'.

Theorems on kernel and homomorphisms

If f:G\to G' with kernel K, then K \lhd G.
Every homomorphic image of a cyclic group is cyclic but not conversely.
Every homomorphic image of a abelian group is abelian but not conversely.
Every group is homomorphic to its quotient group.
- Cor. If p:G\xrightarrow{onto}G/N, then Ker p = N.
Fundamental Theorem of Homomorphism Every homomorphic image of group G is isomorphic to some quotient group of G.

Isomorphic Groups

If \exists f:G\to G' and f is an isomorphism, then G is said to be isomorphic to G' and vice versa i.e. G\cong G'

Theorems on isomorphisms

A homomorphism f defined from a group G onto G' is an isomorphism \iff Ker f=\{e\}.
The relation of isomorphism (\cong) in the set of all groups is an equivalence relation i.e. relation \cong is reflexive, symmetric, and transitive.
- Remark The relation of isomorphism in a family of groups, being an equivalence relation, partitions that family into disjoint equivalence classes. If G\cong G' i.e. if G and G' belong to the same equivalence class, then we simply say that both are isomorphic. When two groups are isomorphic, then their structures will be basically identical. In such case, we some times say that two groups are abstractly identical.
Cayley Theorem Every group is isomorphic to some permutation group i.e. G\cong P_A.
Every infinite cyclic group is isomorphic to the additive group of integers i.e. (\mathbb{Z},+).

Permutation Group (Automorphism)

A permutation f of a finite set S is a bijection from S to itself i.e. f:S\xrightarrow{bijection}S\implies if a\in S then f(a)\in S.

Let S=\{a_1,a_2,\ldots,a_n\} be a finite set of n elements, then permutation group \sigma is \sigma=\left(\begin{array}{cc} a_1 & a_2 & \ldots & a_n \\ f(a_1) & f(a_2) & \ldots & f(a_n) \end{array}\right)

Equality of two permutation

two permutations f and g are equal \iff f(a)=g(a), \forall a\in S i.e. image of every elements of S under both f and g are equal.

Identity permutation

f is identity permutation \iff f(a)=a, \forall a\in S

Product or composition of permutation

Let f and g be permutations of A then product of permutations is also a composition of permutations i.e. (fg)(x)=fog(x)=f[g(x)] - if f and g are not disjoint, they do not commute i.e. fg\neq gf

Cyclic permutation or cycles

\sigma=(a_1\ a_2\ \ldots\ a_n) is a cycle if for a finite subset (a_1, a_2,\ldots, a_n) of S, \sigma(a_1)=a_2, \sigma(a_2)=a_3,\ldots but \sigma(x)\neq x and x\in S\implies x\notin (a_1,a_2,\ldots,a_n)

Length of cycle

No. of elements in the cycle. If length is r, it is called r-cycle. - length of identity permutation cycle is 1.

Order of cycle

(= length of the cycle) No. of times a permutation should be multiplied to get the identity permutation.

Inverse of cycle

If \sigma=(1\ 2\ 3\ 5)\implies \sigma^{-1}=(5\ 3\ 2\ 1)

Disjoint cycle

if two cycles have no common elements - product of two disjoint cycle is commutative - every permutation can be expressed as the product of disjoint cycles

Length of permutation

Order of permutation

O(f)=lcm(length of disjoint cycles)

Transposition

A 2-cycle is called transposition - order of every transposition is 2 - every transposition is self inverse - every permutation is a product of transpositions

Inversion

pair (i,j), 0<i<j\leq n is an inversion for \sigma if \sigma(i)>\sigma(j)

Signature

Sig \sigma = total number of inversions of \sigma

Even and odd permutations

permutation is even (or odd) if the total number of transpositions is even (or odd). - if length of the cycle is even (or odd) \implies permutation is odd (or even) - identity permutation is an even permutation - every transposition is an odd permutation

Product of even (odd) permutation

product of two even permutation is an even permutation
product of two odd permutation is an even permutation
product of one even and one odd permutation is an odd permutation