# Math 4302 Exam 1 Review

> [!NOTE]
>
> This is a review for definitions we covered in the classes. It may serve as a cheat sheet for the exam if you are allowed to use it.

## Groups

### Basic definitions

#### Definition for group

A group is a set $G$ with a binary operation $*$ that satisfies the following axioms:

1. Closure: $\forall a,b\in G, a* b\in G$ (automatically guaranteed by definition of binary operation).
2. Associativity: $\forall a,b,c\in G, (a* b)* c=a* (b* c)$.
3. Identity: $\exists e\in G, \forall a\in G, e* a=a* e=a$.
4. Inverses: $\forall a\in G, \exists a^{-1}\in G, a* a^{-1}=a^{-1}* a=e$.

- Identity element: If $X$ has an identity element, then it is unique.
- Composition of function is associative.

#### Order of a element

The order of an element $a$ in a group $G$ is the size of the smallest subgroup generated by $a$, we denote such subgroup as $\langle a\rangle$.

Equivalently, the order of $a$ is the smallest positive integer $n$ such that $a^n=e$.

#### Order of a group

The order of a group $G$ is the size of $G$.

#### Definition of subgroup

A subgroup $H$ of a group $G$ is a subset of $G$ that is closed under the group operation. Denoted as $H\leq G$.

#### Left and right cosets

If $H$ is a subgroup of $G$, then $aH$ is a coset of $H$ for all $a\in G$. We call $aH$ a left coset of $H$ for $a$. 

$$
aH=\{x|a\sim x\}=\{x\in G|a^{-1}x\in H\}=\{x|x=ah\text{ for some }h\in H\}
$$

Similarly, $Ha$ is a right coset of $H$ for $a$.

$$
Ha=\{x|x\sim'a\}=\{x\in G|xa^{-1}\in H\}=\{x|x=ha\text{ for some }h\in H\}
$$

- Usually, the left coset and right cosets will give different partitions of $G$.
- Use to prove lagrange theorem (partition of $G$ into cosets)

#### Definition of normal subgroup

A subgroup $H$ of a group $G$ is normal if $aH=Ha$ for all $a\in G$.

### Isomorphism and homomorphism

#### Definition of isomorphism

Two groups $G$ and $G'$ are isomorphic if there exists a function $f:G\to G'$ such that

- Homomorphism property is satisfied: $f(a*b)=f(a)f(b),\forall a,b\in G$
- $f$ is injective: $f(a)=f(b)\implies a=b$
- $f$ is surjective: $\forall a\in G',\exists b\in G$ such that $f(b)=a$

#### Definition of homomorphism

A homomorphism is a function that satisfies the homomorphism property.

If $\phi:G\to G'$ is a homomorphism, then

- $\phi(e)=e'$, where $e$ is the identity of $G$ and $e'$ is the identity of $G'$.
- $\phi(a^{-1})=(\phi(a))^{-1}$ for all $a\in G$.
- If $H\leq G$ is a subgroup, then $\phi(H)\leq G'$ is a subgroup.
- If $K\leq G'$ is a subgroup, then $\phi^{-1}(K)\leq G$ is a subgroup.
- $\phi$ is surjective if and only if $\operatorname{ker}(\phi)=\{e\}$ (the trivial subgroup of $G$).

### Basic groups

#### Trivial group

The group $(\{e\},*)$ is called the trivial group.

#### Abelian group

A group $G$ is abelian if $a*b=b*a$ for all $a,b\in G$.

- The smallest non-abelian group is $S_3$ (order 6).
- Every abelian group is isomorphic to some direct product of cyclic groups of the form:
  $$
  \mathbb{Z}_{p_1^{n_1}}\times \mathbb{Z}_{p_2^{n_2}}\times \cdots \times \mathbb{Z}_{p_k^{n_k}}\times\underbrace{\mathbb{Z}\times \ldots \times \mathbb{Z}}_{m\text{ times}}
  $$

#### Cyclic group

A group $G$ is cyclic if $G$ is a subgroup generated by $a\in G$. (may be infinite)

- The smallest non-cyclic group is Klein 4-group (order 4).
- Every group with prime order is cyclic.
- Every cyclic group is abelian.
- If $G$ has order $n$, then $G$ is isomorphic to $(\mathbb{Z}_n,+)$.
- If $G$ is infinite, then $G$ is isomorphic to $(\mathbb{Z},+)$.
- If $G=\langle a\rangle$ and $H=\langle a^k\rangle$, then $|H|=\frac{|G|}{d}$ where $d=\operatorname{gcd}(|G|,|H|)$.
- Every subgroup of cyclic group is also cyclic.

#### Dihedral group

The dihedral group $D_n$ is the group of all symmetries of a regular polygon with $n$ sides.

- $|D_n|=2n$.
- It is finitely generated by $\{\rho,\phi\}$, where $\rho$ is a rotation of a regular polygon by $\frac{2\pi}{n}$, and $\phi$ is a reflection of a regular polygon with respect to $x$-axis.

#### Symmetric group

The symmetric group $S_n$ is the group of all permutations of $n$ objects.

- $S_n$ has order $n!$.
- Every group $G$ is isomorphic to $S_A$ for some $A$.
- Odd and even permutations
  - Every permutation can be written as a product of transpositions.
  - $A_n$ is the alternating group with order $\frac{n!}{2}$ consisting of all even permutations.
  - A non trivial homomorphism from $S_n$ to $(\Z_2,+)$ is given by $\sigma\mapsto \begin{cases} 0 & \text{if } \sigma\text{ is even} \\ 1 & \text{if } \sigma\text{ is odd} \end{cases}$