There exists operator $+$

- $(a+b)+c = a+(b+c)$
- a unity $e \in G$ such that $ae = ea = a$.
- Corresponding to each $a \in G$, there exists inverse $a^{-1}$ of $a$.

such that $aa^{-1}=a^{-1}a=1$

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

A subgroup $K$ of $G$ is **normal** if its left coset and right coset coincide, that is if $gH=Hg$ for all $g \in G$.

Let $H$ be a normal subgroup of $G$. then the cosets of $H$ form a group $G/H$ and is the **factor group**.

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

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

A subgroup $K$ of $G$ is a **conjugate subgroup** oof $H$ if $K=xHx^{-1}$ for some $x \in G$.

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

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

Proof.

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

Sylow p-subgroup $A$ is a maximal p-sbugroup of $G$.

Let $G$ is a finite group and $|G|=p^nm$ where $n \geq 1$ and where $p$ does not divide $m$ then

- $G$ contain a subgroup of order $p^i$ where $1 \leq i \leq n$
- Every subgroup of $H$ of $G$ which has order of $p^i$, is a normal subgroup of order $p^{i+1}$ for $1 \leq i < n$

Let $P_1$ and $P_2$ be sylow p-subgroups. $P_1$ and $P_2$ are conjugate subgroup of $G$.

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

A ring is a nonempty set $R$ with two operations such that

- $(R, +)$ is an abelian group
- $(ab)c=a(bc)$ for all $a, b, c \in R$ (associative multiplication)
- $a(b+c)=ab + ac$ and $(b+c)a=ba + ca$ (left and right distributive laws)

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

A ring which has communtative multiplication is Communtative Ring.

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

and if $R$ contains multiplicative unity $1$, $R$ is said to be **Ring with unity(identity)**.

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

an element $a$ in $R$ is said to be **invertible** or **unit** if there exsits $c \in R$ such that $ac=ca=1_R$.

$R$ is a communtative ring with unity $1_R \neq 0$ and no zero divisor is called a **Integral domain**.

the ring $\mathbb Z$ is an integral domain.

$R$ with nonzero unity in which every nonzero element is a unit is called a **Division ring**.

A field is a communtative division ring.

Let $N$ be a subring of $R$. $N$ satisfying the properties

$r \in R \text{ and } n \in N \quad \Rightarrow \quad rn, nr \in N$

is an **ideal**.

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

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

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

A extension field $E$ of $F$ is the **splitting field over $K$ of the polynomial $f$** if $f$ splits in $F[x]$, that is, if $f = x_0(x - x_1)\cdots(x - x_n)$ with $x_i \in F$, and $F=E(x_0, \ldots, x_n)$.

$E$ is said to be a **seperable extension of $F$** if every elements of $E$ is the simple root $c$ such that $f = (x-c)g(x)$ for $g(c) \neq 0$ and $f \in f[x]$.