# Definitions

**Symmetry**

A rearrangement of a geometric figure preserving the arrangement of its sides and vertices as well as its distances and angles.

**Rigid Motion**

A map from a plane to itself while keeping symmetry of the object.

**Binary Operations**

A binary operation or law of composition on a set $G$ is a function $G \times G \mapsto G$ that assigns to each pair $(a,b) \in G$ a unique element $a \circ b$, or $ab$ in $G$, called the composition of $a$ and $b$.

**Group**

A group $(G, \circ)$ is a set $G$ together with a binary operation $\circ:(a,b) \mapsto a \circ b$ that satisfies:

- $\circ$ is associative: $(a\circ b) \circ c=a\circ (b\circ c); \forall a,b,c\in G$.
- There exists an identity, denoted by $e$ (or $0$ or $1$): $a\circ e = e\circ a = a; \forall a\in G$.
- For each $a \in G$, there exists an inverse, denoted by $a^{-1}$: $a\circ a^{-1}$=$a^{-1}\circ a$=$e$.

**Identity Element**

an element $e\in G$, such that for any element $a\in G$ , $a\circ e = e\circ a = a$

**Abelian Groups**

A group $G$ with the property that $a\circ b = b\circ a$ for all $a,b\in G$ is called abelian or commutative. Groups not satisfying this property are said to be nonabelian or noncummutative.

**Non-Abelian Group**

$S_3$ is an example of a 6 element group that is not abelian. $3!$

$V_4$ is a non-abelian group that is the symmetry group for a non-square rectangle: $\{e, h, v, r\}$

**Subgroup**

A subgroup $H$ of a group $G$ is a subset of $G$ such that the group operation restricted to $H$ ($H$ has the same operation as $G$), $H$ is a subgroup in its own right.

**Proper Subgroup**

If $H\leq G$ and $H\neq G$, then H is called a Proper Subgroup.

**Finite of Finite Order**

A group is ** finite**, or has

**, if it contains a finite number of elements. If the group is not finite then it is said to be infinite or have infinite order.**

*finite order***Order**

The ** order** of a finite group is the number of elements that it contains. The group $G$ containing $n$ elements is written $\mid G \mid = n$.

*Example*- The group $\mathbb{Z}_5$ is a finite group of order 5.

# Theorems

**Proposition 2.1**

Let $n$ be a set of equivalence classes in the integers $mod n$ and $a,b,c\in\mathbb{Z}$.

1. Addition and multiplication can be commutative:

$a+b$ $\equiv$ $b+a$ mod $n$

$ab$ $\equiv$ $ba$ mod $n$

2. Addition and Multiplication can be associative:

$(a+b)+c$ $\equiv$ $a+(b+c)$ mod $n$

$(a*b)$ $c$ $\equiv$ $a$ $(b*c)$

3. The set has an additive identity and multiplicative identity:

$a+0$ $\equiv$ $a$ mod $n$

$a*1$ $\equiv$ $a$ mod $n$

4. Multiplication can distribute over addition:

$a$$(b+c)$ $\equiv$ $a*b+a*c$ mod $n$

5. For every integer $a$ there is an additive inverse $-a$:

$a+(-a)$$\equiv$$0$ mod $n$

**Proposition 2.2**

The identity element in a group $G$ is unique.

Proof. Suppose that $e$ and $e^{l}$ are both identities in G. Then $eg = ge = g$

and $e^{l}g = ge0 = g$ for all $g\in G$. We need to show that $e = e^{l}$. If we think

of $e$ as the identity, then $ee^{l} = e^{l}$; but if $e^{l}$ is the identity, then $ee^{l} = e$.

Combining these two equations, we have $e = ee^{l} = e^{l}$.

inverses in a group are also unique. If $g^{l}$and $g^{ll}$ are both inverses of an

element $g$ in a group G, then $gg^{l} = g^{l}g = e$ and $gg^{ll} = g^{ll}g = e$. We want

to show that $g^{l} = g^{ll}$, but $g^{l} = g^{l}e = g^{l}(gg^{ll}) = (g^{l}g)g^{ll} = eg^{ll} = g^{ll}$. We

summarize this fact in the following proposition.

**Proposition 2.3**

Let $g\in$$G$, then the inverse of g is unique, $g^{-1}$

**Proposition 2.4**

Let $G$ be a group and $a,b\in$$G$, we see that

$(ab)^{-1}=a^{-1}b^{-1}$.

**Proposition 2.5**

Let $G$ be a group. For any $a\in G$, $(a^{-1})^{-1}=a$.

**Proposition 2.6**

Let $G$ be a group and $(a,b)\in$$G$, then

$ax=b$ and $xa=b$ have unique solutions in $G$.

**Proposition 2.7** Cancellation Law for Groups

If $G$ is a group and $a,b \in G$ then $ba=ca$ or $ab=ac$ implies $b=c$.

**Theorem 2.8**

If $G$ is a group and $g\in$$G$, then we define $g^0=e$. For $n\in$$N$,

we define

In a group, the usual laws of exponents hold; that is, for all $g,h\in\G$,

1. $g^m g^n = g^{m+n}$ for all $m,n\in\mathbb{Z}$;

2. $(g^m)^n = g^{mn}$ for all $m,n\in\mathbb{Z}$;

3. $(gh)^n = (h^{-1} g^{-1})^{-n}$ for all $n\in\mathbb{Z}$. Furthermore, if G is abelian, then $(gh)^n = g^n h^n$.

**Proposition 2.9**

A subset $H$ of $G$ is a subgroup iff it satisfies the following conditions:

1. The identity $e$ of $G$ in $H$.

2. If $h_1, H_2 \in H$, then $h_1h_2 \in H$.

3. If $h \in H$, then $h^{-1} \in H$.

**Proposition 2.10**

Let $H$ be a subset of a group $G$. Then $H$ is a subgroup of $G$ iff $H\neq \emptyset$, and whenever $g,h\in H$, $gh^{-1}\in H$.

# Examples & Notes

**Exercise 24**

Let a and b be elements in a group $G$; Prove that $(ab)^{n}(a)^{-1} = (aba^{-1})^{n}$

The statement for n = 1 is simply that $aba^{-1}= a b a^{-1}$, which is certainly true. Now assume that the result holds for $n = k$.

$(aba^{-1})^{k+1} = (aba^{-1})^{k}(a b a^{-1})$

$(a b^{k} a^{-1}) (a b a^{-1})$

$(a b^{k})(a^{-1} a)( b a^{-1} )$

$(a b^{k})(b a^{-1})$

$a b^{k+1} a^{-1}$

Therefore the statement is true for $n=k+1$ which means it holds true for all values of n.

**Exercise 28**

Prove the left and right cancellation laws for a group $G$; that is, show that in the group $G$, $ba = ca$ implies $b = c$ and $ab = ac$ implies $b = c$ for elements $a,b,c\in G$.

Proof:

Let $a,b,c\in G$, and assume $ba = ca$ and $ab = ac$.

We see that:

and

(3)**Claim:** The collection of symmetries of an object forms a group under composition (means doing one symmetry and than another). Example $S_{3} \cong D_{3}$

**Important:** Function composition is associative so composing permutations is associative.

**$S_3$**

**Some Examples of Groups**

1. $\mathbb{Z}$ under +, is a abelian group

2. $\mathbb{Z}$ under multiplication, is not a group, fails inverses

3. $\mathbb{R}$ under + is a abelian group

4. $\mathbb{R}$ under multiplication, is not a group, 0 fails to have an inverse

5. $\mathbb{R*} =\mathbb{R} - {0}$ is a abelian group under multiplication

6. $\mathbb{Z}_{n}$ is a abelian group under + mod $n$

7. $\mathbb{C*} = \mathbb{C} - {0{$ is a group under multiplication

8. $GL_{2} =$ set of all $2 \times 2$ invertible matrices, is a group under matrix multiplication

**Exercise #29**

Show that if $a^{2} = e$ $\forall$ $a \in G$ then G must be an abelian group.

Proof: Let $a,b \in G$ and $a^{2} = e$

Assume: $g^{2} = e$ $\forall$ $g\in G$

We see that:

$abab = (ab)^{2}=e$

$abab = e$

$aabab = ae$

$ebab = a$

$bab = a$

$bbab = ba$

$eab = ba$

$ab = ba$

**Proposition 2.8 pt3 abelian Proof**

Given $G$ is a group, $(ab)^2 = a^2b^2$ for all $a,b \in G$ if $G$ abelian.

Proof:

We know:

$(ab)^2 = a^2b^2$

This means:

$abab = aabb$

Multiply each side on left by $a^{-1}$

$a^{-1}abab =a^{-1}aabb$

The inverse cancels the first $a$'s

$bab = abb$

Now multiply on right by $b^{-1}$

$babb^{-1} = abbb^{-1}$

The right $b's$ cancel

$ba=ab$

Therefore, group $G$ is abelian!!

**Cayley Table for $\mathbb{Z}_{4}$**

**Subgroup lattice of** $Q_8$ (sorry I didn't know how to draw the lines)

**subgroups $D_3$**

$\{{e}\}$, $\{{e,f_3}\}$, $\{{e,f_1}\}$, $\{{e,f_2}\}$, $\{{e, r_1, r_2}\}$, $D_3$

where f represents each of the different flip and $r_1$ is a 120 degree rotation and $r_2$ is a 240 rotation.

ij = k

jk = i

ki = j