MATH 415 Lecture 3

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Groups

(See MATH 415 Lecture 2#Groups→)

More examples:

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

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

Semi-Groups and Monoids

$\left\langle S,*\right\rangle$ is a semi-group if it has only the associative law.

For example, $\left\langle \mathbb {Z} ^{+},+\right\rangle$ is a semi-group because for $a,b\in \mathbb {Z} ^{+}$, $a+b\in \mathbb {Z} ^{+}$, but $-a<0\not \in \mathbb {Z} ^{+}$.

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 $G$:

${\begin{aligned}a\,b=a\,c&\implies b=c\\b\,a=c\,a&\implies b=c\end{aligned}}$

Theorem 4.16

Equations of the form $a\,x=b$ and $y\,a=b$ have unique solutions.

Proof.

${\begin{aligned}a^{-1}\,(a\,x)&=a^{-1}\,((b)\\(a^{-1}\,a)\,x&=a^{-1}\,b\\x&=a^{-1}\,b\\\end{aligned}}$

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 $a,b\in G$, $\left(a\,b\right)^{-1}=b^{-1}\,a^{-1}$

Proof. $(b^{-1}\,a^{-1})\,(a\,b)=b^{-1}\,a^{-1}\,a\,b=e$. 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 $a\neq e$:

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

Group of Three elements, ${-a,e,a}$, where $a\neq e$.

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

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 $\left\langle \mathbb {Z} _{4},+_{4}\right\rangle \not \simeq \left\langle G,*\right\rangle$.

Subgroups

Given a group $\left\langle G,*\right\rangle$ and a group $\left\langle H,*\right\rangle$ is a subgroup of $\left\langle G,*\right\rangle$

By definition of a group, $\left\langle H,*\right\rangle$ satisfies the following properties:

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

Furthermore, $\left\langle H,*\right\rangle$ 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 $\left\langle H,*\right\rangle$ is a "proper subgroup" of $\left\langle G,*\right\rangle$, and $H\leq G$ if $\left\langle H,*\right\rangle$ is just a subgroup of $\left\langle G,*\right\rangle$.

Note: $\left\langle \mathbb {Z} ,+\right\rangle <\left\langle \mathbb {Q} ,+\right\rangle <\left\langle \mathbb {R} ,+\right\rangle <\left\langle \mathbb {C} ,+\right\rangle$

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 $\left\langle \left\{M_{n}(\mathbb {R} )~\mid ~M~{\mbox{is invertible}}\right\},\cdot \right\rangle$ Furthermore, the group of matrices whose determinants is 1 (i.e. $\left\{M_{n}(\mathbb {R} )~\mid ~|M|=1\right\}$) is a subgroup of invertible matrices over $\cdot$.