NoteNextra-origin/content/CSE5313/CSE5313_L5.md

CSE5313 Coding and information theory for data science (Lecture 5)

Recap

Group

1. Closure: \forall a,b\in G, a\cdot b\in G.
2. Associativity: \forall a,b,c\in G, (a\cdot b)\cdot c=a\cdot (b\cdot c).
3. Identity: \exists e\in G, \forall a\in G, a\cdot e=e\cdot a=a.
4. Inverses: \forall a\in G, \exists a^{-1}\in G, a\cdot a^{-1}=a^{-1}\cdot a=e.

May not be commutative (group of invertible matrices).

Order of element in group

a\in G is of order k if a^k=e and k is the smallest positive integer such that a^k=e.

If a^n=e, then O(a)\mid n.

Generator of group

a\in G is a generator of G if \mathcal{O}(a)=|G|. (for finite groups)

For infinite groups, \langle a\rangle=G.

Example

(\mathbb{Z}_n,+) has generator 1.

(\mathbb{Z}_8^*,\cdot) has generator 3. (Recall \mathbb{Z}_8^*=\{x\in\mathbb{Z}_8:gcd(x,8)=1\}. for multiplicative inverse.)

New content

Subgroups

A subgroup H of G is a nonempty subset of G that is itself a group under the operation of G.

Denoted as H\leq G.

Example

(\mathbb{Z}_6,+) has subgroups H=(\{0,2,4\},+).

Only need to check three:

• non-empty
• closure
• finite

Theorem for finite subgroups

If H is finite, non-empty, and closed under the operation of G, then H is a subgroup of G.

Equivalence relations

An equivalence relation \sim on a set X is a relation that is

• reflexive: \forall x\in X, x\sim x
• symmetric: \forall x,y\in X, x\sim y\implies y\sim x
• transitive: \forall x,y,z\in X, x\sim y\text{ and } y\sim z\implies x\sim z

Example

Let S be points on land, and a\sim b if a and b are connected by land.

Equivalence classes

An equivalence relation on S partitions S into equivalence classes.

Equivalence classes are:

• Disjoint
• Cover S

Cosets

Let G be a group and H its subgroup.

The coset of H in G is the equivalence class under congruence modulo H.

Alternatively, (more convenient)

\{h+a|h\in H,a\in G\}

Example

Let G=(\mathbb{Z}_6,+) and H=(\{0,2,4\},+).

Define the equivalence relation on G as:

a\sim b if a+b^{-1}\in H (congruence modulo H)

To find the equivalence classes of this relation for G=(\mathbb{Z}_6,+) and H=(\{0,2,4\},+), we have:

• 0\sim 0, 0\sim 2, 0\sim 4
• 1\sim 1, 1\sim 3, 1\sim 5

Lagrange's theorem

For every finite group G, the order of every subgroup H of G divides the order of G.

Corollary of Lagrange's theorem

If |G| is prime, then G is cyclic.

Proof

Let A=\{a^0=e,a,a^2,\cdots,a^{|G|-1}\}.

A is a cyclic subgroup of G. of order at least two.

Then |A|||G|.

So |A|=|G|.

So A=G.

So G is cyclic.

Additive group of a field

Any finite field has two types groups:

• Additive group: (\mathbb{F},+)
• Multiplicative group: (\mathbb{F}^*,\cdot)

The "integer" of F is:

\{a\in F|1^k=a,k\in\mathbb{N}\}

The "characteristic" of F is:

• The order of 1 in additive group
• Number of times that 1 is added to itself to get 0
• Denoted by \operatorname{c}(F).

Example

\operatorname{c}(\mathbb{Z}_7)=7.

\operatorname{c}(\mathbb{R})=0.

\operatorname{c}(\mathbb{Z}_2[x] \mod x^2+x+1)=2.

Theorem field characteristic is prime

If \operatorname{c}(F)>0, then \operatorname{c}(F) is prime.

Proof

Suppose \operatorname{c}(F)=mn, then 0=\sum_{i=0}^{m-1}1\cdot\sum_{j=0}^{n-1}1=0.

So m or n must be 0.

So \operatorname{c}(F) is prime.

Theorem of linear power over additive group with prime characteristic

Let F be a field with characteristic p>0, then operation ^p is linear.

That is, (a+b)^p=a^p+b^p.

Proof

(a+b)^p=\sum_{i=0}^p \binom{p}{i} a^i b^{p-i}

\begin{aligned}
\binom{p}{i}&=\frac{p!}{i!(p-i)!}\\
&=\frac{p(p-1)\cdots(p-i+1)}{i(i-1)\cdots 1}
\end{aligned}

Informally, p divides the numerator but not the denominator. So the whole fraction is an integer.

Since \binom{p}{i} is an integer of F except for i=0 and i=p, we have \binom{p}{i}=0 for i=1,\cdots,p-1.

So (a+b)^p=a^p+b^p.

Multiplicative group of a field

Every element in a multiplicative group of a field is cyclic.

Corollary:

• Every finite field has a generator, called a primitive element.
• This is an element \gamma such that \mathbb{F}^*=\langle \gamma\rangle.
• Every element of \mathbb{F}^* is a power of \gamma.

Example

Build F_{16}=\mathbb{Z}_2[\zeta] \mod \zeta^4+\zeta+1.

The elements are:

Power of \zeta	Element	As vector in \mathbb{Z}_2^4
0	0	(0,0,0,0)
1	1	(1,0,0,0)
2	\zeta	(0,1,0,0)
3	\zeta^2	(0,0,1,0)
4	\zeta^3	(0,0,0,1)
5	\zeta+1	(1,1,0,0)
6	\zeta^2+\zeta	(0,1,1,0)
7	\zeta^3+\zeta^2	(0,0,1,1)
8	\zeta^3+\zeta^2+1	(1,1,1,0)

The primitive element is \zeta.

Vector spaces and subspaces over finite fields

\mathbb{F}^n is a vector space over \mathbb{F}.

With point-wise vector addition and scalar multiplication.

Example

\mathbb{F}_2^4 is a vector space over \mathbb{F}_2.

Let $v=\begin{pmatrix} 1 & 1 & 1 & 1 \end{pmatrix}$

Then v is a vector in \mathbb{F}_2^4 that's "orthogonal" to itself.

v\cdot v=1+1+1+1=4=0 in \mathbb{F}_2.

In general field, the dual space and space may intersect non-trivially.

Let V be a subspace of \mathbb{F}^n.

V is a subgroup of \mathbb{F}^n under vector addition.

• Apply the theorem: If H is finite, non-empty, and closed under the operation of G, then H is a subgroup of G.

Is every subgroup of \mathbb{F}^n a subspace?

Cosets in this definition are called Affine subspaces.