algebra summary



There exists operator ++

  1. (a+b)+c=a+(b+c)(a+b)+c = a+(b+c)
  2. a unity eGe \in G such that ae=ea=aae = ea = a.
  3. Corresponding to each aGa \in G, there exists inverse a1a^{-1} of aa.
    such that aa1=a1a=1aa^{-1}=a^{-1}a=1

a order of group is a number of elements in a group.

Normal subgroup

A subgroup KK of GG is normal if its left coset and right coset coincide, that is if gH=HggH=Hg for all gGg \in G.

Factor group

Let HH be a normal subgroup of GG. then the cosets of HH form a group G/HG/H and is the factor group.

Simple group

A group is simple if it is nontrivial and has no proper nontrivial normal subgroups.

Cyclic group

Let GG be a group and let aGa \in G. then the subgroup {annZ}\{a^n | n \in \mathbb Z\} this group is the cyclic group a\langle a \rangle of GG generated by aa. and the order a|\langle a \rangle| of this cyclic subgroup called the order of (element) a.

conjugate subgroup

A subgroup KK of GG is a conjugate subgroup oof HH if K=xHx1K=xHx^{-1} for some xGx \in G.

Cauchy theory

Let pp is a prime. Let GG be a finite group and let p divide G|G|. then GG has an element of order pp and consequently a subgroup of order pp.

Langrange theory

For HGH \subset G is a subgroup of GG, H|H| divides G|G|.



Let pp be a prime. A group GG is a p-group if every element in GG has order a power of the prime pp.

Sylow p-subgroup

Sylow p-subgroup AA is a maximal p-sbugroup of GG.

Sylow theory

First sylow theory

Let GG is a finite group and G=pnm|G|=p^nm where n1n \geq 1 and where pp does not divide mm then

  1. GG contain a subgroup of order pip^i where 1in1 \leq i \leq n
  2. Every subgroup of HH of GG which has order of pip^i, is a normal subgroup of order pi+1p^{i+1} for 1i<n1 \leq i < n

Second sylow theory

Let P1P_1 and P2P_2 be sylow p-subgroups. P1P_1 and P2P_2 are conjugate subgroup of GG.

Third sylow theory

If pp divide G|G|, the number of sylow p-subgroups is 1(modp)\equiv 1 (mod\;p) and divide G|G|.



A ring is a nonempty set RR with two operations such that

  1. (R,+)(R, +) is an abelian group
  2. (ab)c=a(bc)(ab)c=a(bc) for all a,b,cRa, b, c \in R (associative multiplication)
  3. a(b+c)=ab+aca(b+c)=ab + ac and (b+c)a=ba+ca(b+c)a=ba + ca (left and right distributive laws)


Let RR be a ring. there exists a least positive number nn such that a1a2an=0a_1 a_2 \ldots a_n = 0, for aiRa_i \in R. RR is said to have characteristic n. if there not exits nn in RR, this called characteristc zero.

Communtative Ring

A ring which has communtative multiplication is Communtative Ring.

ab=ba, for a,b,Rab = ba ,\:\text{ for } a, b, \in R

and if RR contains multiplicative unity 11, RR is said to be Ring with unity(identity).

Zero divisor

A nonzero element aa in a ring RR is said to be a zero divisor if there exists nonzero a,bRa,b \in R such that ab=ba=0ab=ba=0.


an element aa in RR is said to be invertible or unit if there exsits cRc \in R such that ac=ca=1Rac=ca=1_R.

Integral domain

RR is a communtative ring with unity 1R01_R \neq 0 and no zero divisor is called a Integral domain.


the ring Z\mathbb Z is an integral domain.

Division ring

RR with nonzero unity in which every nonzero element is a unit is called a Division ring.

A field is a communtative division ring.



Let NN be a subring of RR. NN satisfying the properties

rR and nNrn,nrNr \in R \text{ and } n \in N \quad \Rightarrow \quad rn, nr \in N

is an ideal.


the cyclic group c\langle c \rangle is an ideal in Z\mathbb Z



If an extension EE of a FF is of finite dimension nn as vector space over FF, then EE is a finite extension of degree nn over FF. let \[E:F\] be the degree nn of EE over FF.


Let EE be a finite extension of a field FF. the number of isomorphism of EE onto subfield of a closure FF leaving FF fixed is the index {E:F}\{E:F\} of EE over FF.

Splitting field

A extension field EE of FF is the splitting field over KK of the polynomial ff if ff splits in F[x]F[x], that is, if f=x0(xx1)(xxn)f = x_0(x - x_1)\cdots(x - x_n) with xiFx_i \in F, and F=E(x0,,xn)F=E(x_0, \ldots, x_n).

Seperable field

EE is said to be a seperable extension of FF if every elements of EE is the simple root cc such that f=(xc)g(x)f = (x-c)g(x) for g(c)0g(c) \neq 0 and ff[x]f \in f[x].