First lecture

[ Home ][ PDF ]

Topics: Groups, Cyclic Groups

Disclaimer: There may be errors her, please report them to me, and if the equations look to terrible, check out the PDF.

Groups we already know

$\mathbb{N}$ = {1,2,3,…} Natural numbers

$\mathbb{Z}$ = {…,-2,-1,0,1,2,…} all integers positive and negative

$\mathbb{Q}$ = { $\frac{p}{q} : p,q \in \mathbb{Z}$ }

$\mathbb{R}$ = {0.1, 0.0 , 0.11, $\sqrt 1, \pi$ }

Relations: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$

Operations

Addidtion $+$ and substraction $-$ Multiplication $\cdot$ divivsion $\div$

$$a-b = a + (-b)$$ $$a \div b = a * \frac{1}{b}$$

Properties

Commutativity

We have for addition $a+b = b + a$ and multiplication $a \cdot b = b \cdot a$ and is called commutativity when $\forall a,b $

Associative

We have for addition $a+(b+c) = (a + b) + c$ and for multiplication $a \cdot(b \cdot c) = (a \cdot b) \cdot c$

Distributivity

$$a ( b + c) = a\cdot b + a\cdot c$$ $$(a + b) c = a\cdot c + b\cdot c$$

Binary Operations

Let $\mathbb{S} $ be a set of elements, with the binary operation $\varphi : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$

Then we have that:

$$\varphi(a,b) = c$$ $$a,b,c \in \mathbb{S}$$

Ternary Operations

$$\varphi : \mathbb{S} \times \mathbb{S} \times \mathbb{S} \to \mathbb{S}$$ $${\varphi(a,b,c) = d}$$ $$a,b,c,d \in \mathbb{S}$$

This can be extended to a n-ary operation

$$\varphi : \mathbb{S_1} \times \mathbb{S_2} \times ... \times \mathbb{S_n}$$ $$n \in \mathbb{N}$$

Algebraic System / Structure $<S,P>$

Algebraic system with the symbol $*$ as a binary operation. Here $S$ is called the groupoid.

$$<S, * >$$

$S$ a groupoid where the assosiative laws holds, is called a semigroupoid.

$$a \times (b \times c) = (a \times b) \times c$$ $$\forall a,b,c$$

$S$ is a semigroup with $e \in S$ s.t $e\times a = a \times e = a, \forall a$ is called a monoide.

$e$ is called an identity element.

If $S$ is a monoid and $ \forall a \in S , \exists a^{-1} \in S$ s.t $ a \times a^{-1} \wedge a^{-1} \times a = e$ Then $S$ is called a group.

$a^{-1}$ is called the inverse element of $a$.

Definition of a group

$\langle G, * \rangle$ with a binary operation $*$, is called a group if the following holds:

i) $\forall a,b,c \in G$ we have that $a * (b * c) = (a * b) * c$

ii) $\forall a\in G, \exists e\in G$ we have that $e * a = a * e = a$

iii) $\forall a \in G, \exists a^{-1}$ such that $a*a^{-1} = a^{-1} * a = e$

If a groups is also commutative, then it is called an abelian group. Then this also applies to the group.

iv) $a*b = b * a$

Proof: Identity element of a group is unique

The identity element $e$ of a group $G$ is unique.

$$\exists e_1,e_2 \in G$$ $$e_1 \times a = a \times e_1$$ $$e_2 \times a = a \times e_2$$ $$\implies e_1 = e_1 \times e_2 = e_2$$ $$\square$$

The inverse element is unique $\forall a \in G$ in group $G$.

Proof: Inverse element $\forall a \in G$ is unique

Let $a$ be an element in $G$, $a\in G$,

Assume we have to inverse elements $a^{-1}$ and $a_1^{-1}$, for $a^{-1}$ and $a_1^{-1}$ the following holds.

$$(a^{-1}\times a = a \times a^{-1} = e$$ $$a_1^{-1} \times a = a \times a_1^{-1} = e$$

To show that $a$ only has one inverse element.

$$a^{-1} \times a = e$$ $$(a^{-1} \times a) \times a_1^{-1} = e \times a_1^{-1}$$ $$a^{-1} \times (a \times a_1^{-1}) = a_1^{-1}$$ $$a^{-1} \times e = a_1^{-1}$$ $$a^{-1} = a_1^{-1}$$ $$\square$$

If there is two inverse elements, they are the same element.

The identity $e$ of a group $G$ is unique, The inverse element is unique $\forall a \in G$ in group $G$

Then $\forall a,b$ we have the following

$$(a \times b)^{-1} = b^{-1} \times a^{-1}$$

We can then show that:

$$(a \times b)^{-1} \times (a \times b) = e$$ $$(b^{-1} \times a^{-1}) \times (a \times b) = b^{-1} \times (a^{-1} \times a ) \times b^{-1}$$ $$RHS = b^{-1} \times e \times b$$ $$= b^{-1} \times b$$ $$(b^{-1} \times a^{-1}) \times (a \times b) = e$$ $$\square$$

$*$ Operator

The $*$ operator will be replaced by either $+$ or $\cdot$ for their repective operation, $\cdot$ => multiplication, and $+$ for addition.

Multiplicative notation

$$* := \cdot$$ $$e = 1 \text{ (the identity element)}$$ $$a^{-1} \text{ (the inverse element)}$$ $$a \cdot b = ab$$ $$a_1a_2....a_n$$ $$a\cdot a \cdot ... \cdot a = a^n$$ $$(-a)\cdot (-a) \cdot ... \cdot (-a) = (-a)^n$$ $$(-a^{-1})\cdot (-a^{-1}) \cdot ... \cdot (-a^{-1}) = (-a)^{-n}$$ $$a^0 = e$$ $$a^n , n \in \mathbb{Z}$$ $$\forall n, m \in \mathbb{Z} \text{ we haven } a^n \cdot a^m + a^{a + m}$$ $$(a^n)^m = a^{n\cdot m}$$ $$n\cdot (m \cdot a) = (n\cdot m)\cdot a$$

Additive notation

$$*:= +$$ $$e = 0 \text{ (the identity element)}$$ $$-a \text{ (the inverse element)}$$ $$0 \cdot a = 0$$ $$a_1 + a_2 + ... + a_n$$ $$a_1+a_2 + ... + a_n = n \cdot a$$ $$n \cdot a + m \cdot a = (n + m)\cdot a$$

Example: $\mathbb{Z}$

$\langle \mathbb{Z}, +\rangle$ is a group, it must fulfil the group definition.

i) $\forall a,b,c \in \mathbb{Z}$ we have $ a + (b + c) = (a + b) + c$ //OK

ii) $e = 0$ for addetive groups $ a + e = a \iff e = 0$ //OK

iii) If $a$ is an element in $\mathbb{Z}$, $a \in \mathbb{Z}$ then it has an inverse s.t the following holds:

$$a^{-1} + a = e$$ $$(-a) + a = 0$$

All tree rules holds, it is a group.

Example: Trivial Group

In a trivial group, the identity element must exists.

$$G = \{ e \}$$ $$e * e = e$$

Example: $\mathbb{Q}$

Is $\langle \mathbb{Q}, +\rangle$ a group? All three rules holds for this group, so yeas this is a group.

Is $\langle \mathbb{Q}, \cdot\rangle$ a group? Associative property holds Identity property $e = 1$ for multiplicative groups.

Inverse Element: For $\mathbb{Q}$ we have that $a = \frac{p}{p}$ then the inverse element is $a^{-1} = \frac{q}{p}$. If $p = 0$ then $a^{-1}$ is not a number ( $p = 0 \rightarrow \frac{q}{0} = NaN $), and hence $\langle\mathbb{Q},\cdot\rangle$ cannot be a group.

Example: G with 6 elements

Let $G$ be the set of remainders of all the integers on division by 6, this group contains of 6 elements and is:

$$G = \{ 0, 1, 2, 3, 4, 5 \}$$

Not sure why this got noted:

An element $m\in \mathbb{Z}$ is defined as follows:

$m = g \cdot q + r$ where $q \in \mathbb{Z} \text{ and } 0 \leq r \leq 5$