# Math 4302 Exam 2 Review

## Groups


### Direct products

$\mathbb{Z}_m\times \mathbb{Z}_n$ is cyclic if and only if $m$ and $n$ have greatest common divisor $1$.

More generally, for $\mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}\times \cdots \times \mathbb{Z}_{n_k}$, if $n_1,n_2,\cdots,n_k$ are pairwise coprime, then the direct product is cyclic.


If $n=p_1^{m_1}\ldots p_k^{m_k}$, where $p_i$ are distinct primes, then the group 

$$
G=\mathbb{Z}_n=\mathbb{Z}_{p_1^{m_1}}\times \mathbb{Z}_{p_2^{m_2}}\times \cdots \times \mathbb{Z}_{p_k^{m_k}}
$$

is cyclic.

### Structure of finitely generated abelian groups

#### Theorem for finitely generated abelian groups

Every finitely generated abelian group $G$ is isomorphic to 

$$
Z_{p_1}^{n_1}\times Z_{p_2}^{n_2}\times \cdots \times Z_{p_k}^{n_k}\times\underbrace{\mathbb{Z}\times \ldots \times \mathbb{Z}}_{m\text{ times}}
$$

#### Corollary for divisor size of abelian subgroup

If $g$ is abelian and $|G|=n$, then for every divisor $m$ of $n$, $G$ has a subgroup of order $m$.

> [!WARNING]
>
> This is not true if $G$ is not abelian.
>
> Consider $A_4$ (alternating group for $S_4$) does not have a subgroup of order 6.


### Cosets

#### Definition of Cosets

Let $G$ be a group and $H$ its subgroup.

Define a relation on $G$ and $a\sim b$ if $a^{-1}b\in H$.

This is an equivalence relation.

- Reflexive: $a\sim a$: $a^{-1}a=e\in H$
- Symmetric: $a\sim b\Rightarrow b\sim a$: $a^{-1}b\in H$, $(a^{-1}b)^{-1}=b^{-1}a\in H$
- Transitive: $a\sim b$ and $b\sim c\Rightarrow a\sim c$ : $a^{-1}b\in H, b^{-1}c\in H$, therefore their product is also in $H$, $(a^{-1}b)(b^{-1}c)=a^{-1}c\in H$

So we get a partition of $G$ to equivalence classes.

Let $a\in G$, the equivalence class containing $a$

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

This is called the coset of $a$ in $H$.

#### Definition of Equivalence Class

Let $a\in H$, and the equivalence class containing $a$ is defined as:

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

#### Properties of Equivalence Class

$aH=bH$ if and only if $a\sim b$.

#### Lemma for size of cosets

Any coset of $H$ has the same cardinality as $H$.

Define $\phi:H\to aH$ by $\phi(h)=ah$.

$\phi$ is an bijection, if $ah=ah'\implies h=h'$, it is onto by definition of $aH$.

#### Corollary: Lagrange's Theorem

If $G$ is a finite group, and $H\leq G$, then $|H|\big\vert |G|$. (size of $H$ divides size of $G$)

### Normal Subgroups

#### Definition of Normal Subgroup

A subgroup $H\leq G$ is called a normal subgroup if $aH=Ha$ for all $a\in G$. We denote it by $H\trianglelefteq G$


#### Lemma for equivalent definition of normal subgroup

The following are equivalent:

1. $H\trianglelefteq G$
2. $aHa^{-1}=H$ for all $a\in G$
3. $aHa^{-1}\subseteq H$ for all $a\in G$, that is $aha^{-1}\in H$ for all $a\in G$

### Factor group

Consider the operation on the set of left coset of $G$, denoted by $S$. Define

$$
(aH)(bH)=abH
$$

#### Condition for operation

The operation above is well defined if and only if $H\trianglelefteq G$.

#### Definition of factor (quotient) group

If $H\trianglelefteq G$, then the set of cosets with operation:

$$
(aH)(bH)=abH
$$

is a group denoted by $G/H$. This group is called the quotient group (or factor group) of $G$ by $H$.

#### Fundamental homomorphism theorem (first isomorphism theorem)

If $\phi:G\to G'$ is a homomorphism, then the function $f:G/\ker(\phi)\to \phi(G)$, ($\phi(G)\subseteq G'$) given by $f(a\ker(\phi))=\phi(a)$, $\forall a\in G$, is an well-defined isomorphism.

> - If $G$ is abelian, $N\leq G$, then $G/N$ is abelian.
> - If $G$ is finitely generated and $N\trianglelefteq G$, then $G/N$ is finitely generated.

#### Definition of simple group

$G$ is simple if $G$ has no proper ($H\neq G,\{e\}$), normal subgroup.

### Center of a group

Recall from previous lecture, the center of a group $G$ is the subgroup of $G$ that contains all elements that commute with all elements in $G$.

$$
Z(G)=\{a\in G\mid \forall g\in G, ag=ga\}
$$

this subgroup is normal and measure the "abelian" for a group.

#### Definition of the commutator of a group

Let $G$ be a group and $a,b\in G$, the commutator $[a,b]$ is defined as $aba^{-1}b^{-1}$.

$[a,b]=e$ if and only if $a$ and $b$ commute.

Some additional properties:

- $[a,b]^{-1}=[b,a]$

#### Definition of commutator subgroup

Let $G'$ be the subgroup of $G$ generated by all commutators of $G$.

$$
G'=\{[a_1,b_1][a_2,b_2]\ldots[a_n,b_n]\mid a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n\in G\}
$$

Then $G'$ is the subgroup of $G$.

- Identity: $[e,e]=e$
- Inverse: $([a_1,b_1],\ldots,[a_n,b_n])^{-1}=[b_n,a_n],\ldots,[b_1,a_1]$

Some additional properties:

- $G$ is abelian if and only if $G'=\{e\}$
- $G'\trianglelefteq G$
- $G/G'$ is abelian
- If $N$ is a normal subgroup of $G$, and $G/N$ is abelian, then $G'\leq N$.

### Group acting on a set

#### Definition for group acting on a set

Let $G$ be a group, $X$ be a set, $X$ is a $G$-set or $G$ acts on $X$ if there is a map

$$
G\times X\to X
$$
$$
(g,x)\mapsto g\cdot x\, (\text{ or simply }g(x))
$$

such that

1. $e\cdot x=x,\forall x\in X$
2. $g_2\cdot(g_1\cdot x)=(g_2 g_1)\cdot x$

#### Group action is a homomorphism

Let $X$ be a $G$-set, $g\in G$, then the function

$$
\sigma_g:X\to X,x\mapsto g\cdot x
$$

is a bijection, and the function $\phi:G\to S_X, g\mapsto \sigma_g$ is a group homomorphism.


#### Definition of orbits

We define the equivalence relation on $X$

$$
x\sim y\iff y=g\cdot x\text{ for some }g
$$

So we get a partition of $X$ into equivalence classes: orbits

$$
Gx\coloneqq \{g\cdot x|g\in G\}=\{y\in X|x\sim y\}
$$

is the orbit of $X$.

$x,y\in X$ either $Gx=Gy$ or $Gx\cap Gy=\emptyset$.

$X=\bigcup_{x\in X}Gx$.

#### Definition of isotropy subgroup

Let $X$ be a $G$-set, the stabilizer (or isotropy subgroup) corresponding to $x\in X$ is

$$
G_x=\{g\in G|g\cdot x=x\}
$$

$G_x$ is a subgroup of $G$. $G_x\leq G$.

- $e\cdot x=x$, so $e\in G_x$
- If $g_1,g_2\in G_x$, then $(g_1g_2)\cdot x=g_1\cdot(g_2\cdot x)=g_1 \cdot x$, so $g_1g_2\in G_x$
- If $g\in G_x$, then $g^{-1}\cdot g=x=g^{-1}\cdot x$, so $g^{-1}\in G_x$

#### Orbit-stabilizer theorem

If $X$ is a $G$-set and $x\in X$, then

$$
|Gx|=(G:G_x)=\text{ number of left cosets of }G_x=\frac{|G|}{|G_x|}
$$

#### Theorem for orbit with prime power groups

Suppose $X$ is a $G$-set, and $|G|=p^n$ for some prime $p$. Let $X_G$ be the set of all elements in $X$ whose orbit has size $1$. (Recall the orbit divides $X$ into disjoint partitions.) Then $|X|\equiv |X_G|\mod p$.

#### Corollary: Cauchy's theorem

If $p$ is prime and $p|(|G|)$, then $G$ has a subgroup of order $p$.

> This does not hold when $p$ is not prime.
>
> Consider $A_4$ with order $12$, and $A_4$ has no subgroup of order $6$.

#### Corollary: Center of prime power group is non-trivial

If $|G|=p^m$, then $Z(G)$ is non-trivial. ($Z(G)\neq \{e\}$)

#### Proposition: Prime square group is abelian

If $|G|=p^2$, where $p$ is a prime, then $G$ is abelian.


### Classification of small order

Let $G$ be a group

- $|G|=1$
  - $G=\{e\}$
- $|G|=2$
  - $G\simeq\mathbb{Z}_2$ (prime order)
- $|G|=3$
  - $G\simeq\mathbb{Z}_3$ (prime order)
- $|G|=4$
  - $G\simeq\mathbb{Z}_2\times \mathbb{Z}_2$
  - $G\simeq\mathbb{Z}_4$
- $|G|=5$
  - $G\simeq\mathbb{Z}_5$ (prime order)
- $|G|=6$
  - $G\simeq S_3$
  - $G\simeq\mathbb{Z}_3\times \mathbb{Z}_2\simeq \mathbb{Z}_6$
<details>
<summary>Proof</summary>

$|G|$ has an element of order $2$, namely $b$, and an element of order $3$, namely $a$.

So $e,a,a^2,b,ba,ba^2$ are distinct.

Therefore, there are only two possibilities for value of $ab$. ($a,a^2$ are inverse of each other, $b$ is inverse of itself.)

If $ab=ba$, then $G$ is abelian, then $G\simeq \mathbb{Z}_2\times \mathbb{Z}_3$.

If $ab=ba^2$, then $G\simeq S_3$.
</details>

- $|G|=7$
  - $G\simeq\mathbb{Z}_7$ (prime order)
- $|G|=8$
  - $G\simeq\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2$
  - $G\simeq\mathbb{Z}_4\times \mathbb{Z}_2$
  - $G\simeq\mathbb{Z}_8$
  - $G\simeq D_4$
  - $G\simeq$ quaternion group $\{e,i,j,k,-1,-i,-j,-k\}$ where $i^2=j^2=k^2=-1$, $(-1)^2=1$. $ij=l$, $jk=i$, $ki=j$, $ji=-k$, $kj=-i$, $ik=-j$.
- $|G|=9$
  - $G\simeq\mathbb{Z}_3\times \mathbb{Z}_3$
  - $G\simeq\mathbb{Z}_9$ (apply the corollary, $9=3^2$, these are all the possible cases)
- $|G|=10$
  - $G\simeq\mathbb{Z}_5\times \mathbb{Z}_2\simeq \mathbb{Z}_{10}$
  - $G\simeq D_5$
- $|G|=11$
  - $G\simeq\mathbb{Z}_11$ (prime order)
- $|G|=12$
  - $G\simeq\mathbb{Z}_3\times \mathbb{Z}_4$
  - $G\simeq\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_3$
  - $A_4$
  - $D_6\simeq S_3\times \mathbb{Z}_2$
  - ??? One more
- $|G|=13$
  - $G\simeq\mathbb{Z}_{13}$ (prime order)
- $|G|=14$
  - $G\simeq\mathbb{Z}_2\times \mathbb{Z}_7$
  - $G\simeq D_7$


#### Lemma for group of order $2p$ where $p$ is prime

If $p$ is prime, $p\neq 2$, and $|G|=2p$, then $G$ is either abelian $\simeq \mathbb{Z}_2\times \mathbb{Z}_p$ or $G\simeq D_p$

## Ring


### Definition of ring

A ring is a set $R$ with binary operation $+$ and $\cdot$ such that:

- $(R,+)$ is an abelian group.
- Multiplication is associative: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
- Distribution property: $a\cdot (b+c)=a\cdot b+a\cdot c$, $(b+c)\cdot a=b\cdot a+c\cdot a$. (Note that $\cdot$ may not be abelian, may not even be a group, therefore we need to distribute on both sides.)

> [!NOTE]
>
> $a\cdot b=ab$ will be used for the rest of the sections.

#### Properties of rings

Let $0$ denote the identity of addition of $R$. $-a$ denote the additive inverse of $a$.

- $0\cdot a=a\cdot 0=0$
- $(-a)b=a(-b)=-(ab)$, $\forall a,b\in R$
- $(-a)(-b)=ab$, $\forall a,b\in R$

#### Definition of commutative ring

A ring $(R,+,\cdot)$ is commutative if $a\cdot b=b\cdot a$, $\forall a,b\in R$.

#### Definition of unity element

A ring $R$ has unity element if there is an element $1\in R$ such that $a\cdot 1=1\cdot a=a$, $\forall a\in R$.

#### Definition of unit

Suppose $R$ is a ring with unity element. An element $a\in R$ is called a unit if there is $b\in R$ such that $a\cdot b=b\cdot a=1$.

In this case $b$ is called the inverse of $a$.

#### Definition of division ring

If every $a\neq 0$ in $R$ has a multiplicative inverse (is a unit), then $R$ is called a division ring.

#### Definition of field

A commutative division ring is called a field.

#### Units in $\mathbb{Z}_n$ is coprime to $n$

More generally, $[m]\in \mathbb{Z}_n$ is a unit if and only if $\operatorname{gcd}(m,n)=1$.

### Integral Domains

#### Definition of zero divisors

If $a,b\in R$ with $a,b\neq 0$ and $ab=0$, then $a,b$ are called zero divisors.

#### Zero divisors in $\mathbb{Z}_n$

$[m]\in \mathbb{Z}_n$ is a zero divisor if and only if $\operatorname{gcd}(m,n)>1$ ($m$ is not a unit).

#### Corollaries of integral domain

If $R$ is a integral domain, then we have cancellation property $ab=ac,a\neq 0\implies b=c$.

#### Units with multiplication forms a group

If $R$ is a ring with unity, then the units in $R$ forms a group under multiplication.

### Fermat’s and Euler’s Theorems

#### Fermat’s little theorem

If $p$ is not a divisor of $m$, then $m^{p-1}\equiv 1\mod p$.

#### Corollary of Fermat’s little theorem

If $m\in \mathbb{Z}$, then $m^p\equiv m\mod p$.

#### Euler’s totient function

Consider $\mathbb{Z}_6$, by definition for the group of units, $\mathbb{Z}_6^*=\{1,5\}$.

$$
\phi(n)=|\mathbb{Z}_n^*|=|\{1\leq x\leq n:gcd(x,n)=1\}|
$$

#### Euler’s Theorem

If $m\in \mathbb{Z}$, and $gcd(m,n)=1$, then $m^{\phi(n)}\equiv 1\mod n$.

#### Theorem for existence of solution of modular equations

$ax\equiv b\mod n$ has a solution if and only if $d=\operatorname{gcd}(a,n)|b$ And if there is a solution, then there are exactly $d$ solutions in $\mathbb{Z}_n$.

### Ring homomorphisms

#### Definition of ring homomorphism

Let $R,S$ be two rings, $f:R\to S$ is a ring homomorphism if $\forall a,b\in R$,

- $f(a+b)=f(a)+f(b)\implies f(0)=0, f(-a)=-f(a)$
- $f(ab)=f(a)f(b)$

#### Definition of ring isomorphism

If $f$ is a ring homomorphism and a bijection, then $f$ is called a ring isomorphism.