MATH 415 Lecture 3

« previous | Tuesday, September 3, 2013 | next »

Groups

(See MATH 415 Lecture 2#Groups→)

More examples:

Let $X^{*}=X-\{0\}$

$({\mathbb{\mathbb{Q}}}^{*},\cdot )$, $({\mathbb{ℝ}}^{*},\cdot ),({\mathbb{ℂ}}^{*},\cdot )$ are all commutative (i.e. abelian) groups.

Semi-Groups and Monoids

$⟨S,*⟩$ is a semi-group if it has only the associative law.

For example, $⟨{\mathbb{\mathbb{Z}}}^{+},+⟩$ is a semi-group because for $a,b\in {\mathbb{\mathbb{Z}}}^{+}$, $a+b\in {\mathbb{\mathbb{Z}}}^{+}$, but $-a<0\in ̸{\mathbb{\mathbb{Z}}}^{+}$.

A semi-group with an identity element is called a Monoid.

Elementary Properties

Theorem 4.15

If $G$ is a group, then the left and right cancellatian laws hold in $G$:

$\begin{array}{rl}ab=ac& ⟹b=c\\ ba=ca& ⟹b=c\end{array}$

Theorem 4.16

Equations of the form $ax=b$ and $ya=b$ have unique solutions.

Proof.

$\begin{array}{rl}{a}^{-1}(ax)& =a^{-1}((b)\\ (a^{-1}a)x& =a^{-1}b\\ x& =a^{-1}b\\ \end{array}$

Similarly, $y=b{a}^{-1}$

quod erat demonstrandum

Theorem 4.17

The entity element is unique

Proof. Suppose there are two,

$e_1$

and

$e_2$

. Then

$e_1{e}_2=e_1$

and

$e_1{e}_2=e_2$

. Thus

$e_1=e_2$

.

quod erat demonstrandum

Corollary

For any two elements $a,b\in G$, ${\left(ab\right)}^{-1}=b^{-1}{a}^{-1}$

Proof. $(b^{-1}{a}^{-1})(ab)=b^{-1}{a}^{-1}ab=e$. In particular, $a{a}^{-1}=e$ and $b{b}^{-1}=e$. This can be extended for an arbitrary number of parenthesized elements by induction.

quod erat demonstrandum

Cardinality

A very important characteristic of a group is the number of elements, denoted $|G|$. This is also called the order of the group.

A group must have at least one element $e$. Group of two elements, $a,e$ where $a\ne e$:

$*$	$e$	$a$
$e$	$e$	$a$
$a$	$a$	$e$

Group of Three elements, $-a,e,a$, where $a\ne e$.

$*$	$e$	$a$	$b$
$e$	$e$	$a$	$b$
$a$	$a$	$b$	$e$
$b$	$b$	$e$	$a$

For arbitrary order, we know that each row and each column must contain all elements of $G$ because of theorem 4.16. Hence the positioning of elements in the table for a group of order 3.

Now things get interesting: there are two possibilites for a group of order 4:

$*$	$e$	$a$	$b$	$c$
$e$	$e$	$a$	$b$	$c$
$a$	$a$	$e$	$c$	$b$
$b$	$b$	$c$	$e$	$a$
$c$	$c$	$b$	$a$	$e$

This is called a Klein Group.

Note that $⟨{\mathbb{\mathbb{Z}}}_4,{+}_4⟩\simeq ̸⟨G,*⟩$.

Subgroups

Given a group $⟨G,*⟩$ and a group $⟨H,*⟩$ is a subgroup of $⟨G,*⟩$

By definition of a group, $⟨H,*⟩$ satisfies the following properties:

• $g,h\in H$ implies $g*h\in H$
• $e_H\in H$ is the identity element
• $\mathrm{\forall }h\in H\mathrm{\exists }{h}^{-1}\in H(h{h}^{-1}=h^{-1}h=e_H)$

Furthermore, $⟨H,*⟩$ must also satisfy the following properties:

• $H\subseteq G$ and is closed under same binary operation $*$
• The identity element $e$ of $G$ is in $H$.
• For all $a\in H$, it is true that $a^{-1}\in H$ also

Notation

We say $H<G$ if $⟨H,*⟩$ is a "proper subgroup" of $⟨G,*⟩$, and $H\le G$ if $⟨H,*⟩$ is just a subgroup of $⟨G,*⟩$.

Note: $⟨\mathbb{\mathbb{Z}},+⟩<⟨\mathbb{\mathbb{Q}},+⟩<⟨\mathbb{ℝ},+⟩<⟨\mathbb{ℂ},+⟩$

Trivial Subgroup

The trivial subgroup $\left\{e\right\}$ is a subgroup of every group.

For example, the identity matrix is a trivial subgroup of the group $⟨\{M_n(\mathbb{ℝ})\mid M\text{is invertible}\},\cdot ⟩$ Furthermore, the group of matrices whose determinants is 1 (i.e. $\{M_n(\mathbb{ℝ})\mid |M|=1\}$) is a subgroup of invertible matrices over $\cdot$.