1. Groups

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Groups are algebraic objects, and it is the study of these objects that forms a central part of abstract algebra, Group Theory. Before we get into our definition of a group, I will first introduce a few definitions that we will use to ultimately define a group.

Before we begin, you will need to remember that the symbol; $\forall$ means “for all elements”, $\exists$ means “there exists an element”, $\in$ means “in the set” and $\mathbb{Z}$ denotes the set of integers (whole numbers).

Definition 1.1: Sets, Elements, Non-Empty and the Empty Set

A Set is a collection of objects. The objects that make up a set are called the Elements of the set. A set is Non-Empty if it has one or more elements. The Empty Set denoted $\emptyset$ is the unique set with no elements.

Example 1.1: $X=\{1,2,3,4,5\}$ is a set and 1, 2, 3, 4 and 5 are its elements.

Definition 1.2: Cardinality

The Cardinality or size of a finite set A is the number of elements in the set A, denoted |a|. (we will look at the cardinality of infinite sets such as the Integers in future posts).

Example 1.2: For $X=\{1,2,3,4,5\}$ we have |X|=5, i.e. the set X has 5 elements.

Definition 1.3: Subset

Let A and B be sets. A is a Subset of B if every element of A is an element of B. This is denoted by A $\subseteq$ B.

Example 1.3: The set $Y=\{2,4\}$ is a subset of $X=\{1,2,3,4,5\}$ .

Definition 1.4: Composition Law

A Composition law (or Binary Function) on a set A is a rule $(\star)$ assigning to each $x\in A$ and $y\in A$ an element $z\in A$ such that $z=x\star y$ . We say that the set is closed

Example 1.4: Addition, +, on the set of all Integers, $\mathbb{Z}$ , is a compositon law. 1+2=3, we have assigned z=3 to x=1 and y=2.

Definition 1.5: Associative

A composition law $(\star)$ on a set A is associative if $(a \star b) \star c = a \star (b \star c), \forall a,b,c\in A$

Example 1.5: Addition on the set of all integers is Associative. e.g. (1+2)+3=3+3=6=1+5=1+(2+3)

We can write this more generally as follows: $\forall a,b,c \in \mathbb{Z}$ we have $(a+b)+c=a+(b+c)$ .

Definition 1.6: Commutative

A composition law $(\star)$ on a set A is commutative if $x\star y=y \star x, \forall x,y\in A$ .

Example 1.6: Addition on the Integers is Commutative. e.g 1+2=3=2+1.

More generally; $\forall a,b \in \mathbb{Z}$ we have $a+b=b+a$ .

Definition 1.7: Group

A pair $(G,\star)$ , where G is a non-empty set and $(\star)$ is a composition law, is a group if:

( $\star$ ) is associative: $a\star (b\star c)=(a\star b)\star c, \forall a,b,c \in \mathbb{Z}$ .
( $\star$ ) admits a neutral element: $\exists e_{G}\in G$ such that $e_{G}\star a=a\star e_{G}=a, \forall a\in G$
Every element of G has an inverse element in G: $\forall a\in G, \exists a^{-1}$ such that $a\star a^{-1}=a^{-1}\star a=e_{G}$

Example 1.7: The Integers, $\mathbb{Z}$ , endowed with addition, +, admits a neutral element, namely 0. e.g. 1+0=0+1=1.

More generally; $\forall a\in \mathbb{Z}, a+0=0+a=a$ .

Example 1.8: Every element of the Integers has an inverse element in the Integers with respect to the composition law of addition. e.g. 4+(-4)=(-4)+4=0.

More generally; $\forall a\in \mathbb{Z}, \exists (-a)$ such that $a+(-a)=0$ .

Now if we take a step back and look at examples 1.4, 1.5, 1.6, 1.7 and 1.8 all together we can see that the Integers endowed with Addition forms a group, which we denote by $(\mathbb{Z},+)$ .

Definition 1.8: Abelian

A Group is Abelian if its composition law is commutative.

Indeed the Integers endowed with Addition forms an Abelian Group.

Properties: Group

The neutral element, $e$ , is unique.
The inverse of an element is unique.
$(a^{-1})^{-1}=a$ .
$(ab)^{-1}=b^{-1}a^{-1}$ .
Cancellation laws: $au=av \Rightarrow u=v$ and $ub=vb \Rightarrow u=v$ .
$a^{n}a^{m}=a^{n+m}$ and $(a^{n})^{m}=a^{nm}$ .

For any element a of a group we can form a subgroup $<a>=\{a^{n} : n\in \mathbb{Z} \}$ for all integer powers of a.

Definition 1.9: Order of a Group

The Order of a group G is the cardinality of the set, denoted |G|.

2. Dihedral Groups and Subgroups

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I will now introduce a family of groups related to the symmetries of polygons, namely Dihedral Groups. We will focus on the Dihedral group related to the symmetries of a square. Consider all the actions you can do to a square to put it back on its self. We can reflect the square along one of its axes of symmetry which we will denote by $S_{x}$ where $x\in\{e,v,d,h\}$ is the axis of symmetry as shown in the following diagram;

We can also rotate the square by multiples of 90 degrees. We let r denote a 90 degree clockwise rotation, $r^{2}$ denotes 2 of these rotations (i.e a 180 degree rotation) and $r^{3}$ denotes a 270 degree clockwise rotation (or equivalently a 90 degree anticlockwise rotation). Putting all of these elements together with the identity element (the action of nothing on the square) with the commutative binary operation of composition (applying the first one and then the second) forms the dihedral group denoted by $D_{8}=\{e, r, r^{2}, r^{3}, s_{h}, s_{v}, s_{d}, s_{e}\}$ , in the following diagram we allocate each corner a colour so that we can see how it moves with each of the elements. The dihedral groups are denoted by $D_{2n}$ where n is the number of sides of the polygon the group is related to the symmetries of.

Figure 2: Visual representation of the Dihedral Group $D_{8}$

Figure 2: Visual representation of the Dihedral Group $D_{8}$

We only go up to $r^{3}$ because $r^{4}=e$ . Take a second to convince yourself that it is closed under the composition law (composing two elements gives another element of the group), especially with respect to the reflections in the axes of symmetry.

Definition 2.1: Order of an element

Let G be a group and $a \in G$ , the order of a is $ord(a)=min\{n\geq 1;a^{n}=e_{G}\}$ if there exists $n \geq 1$ with $a^{n}=e_{G}$ . Otherwise $ord(a)=\infty$ .

Here $ord(a)=min\{n\geq 1;a^{n}=e_{G}\}$ means the smallest number that is greater than or equal to 1 such that $a^{n}=e_{G}$ .

In terms of $D_{8}$ the order of an element is more informally the number of times you can compose the element with itself to get the original square i.e. the identity element e.

In any group, the identity is the unique element of order 1.

Example 2.1: Let us examine the elements of $D_{8}=\{e, r, r^{2}, r^{3}, s_{h}, s_{v}, s_{d}, s_{e}\}$ , take a second to convince yourself of these result;

$ord(e)=1$
$ord(r)=4$
$ord(r^{2})=2$
$ord(r^{3})=4$
$ord(s_{x})=2, \forall x\in\{h,v,d,e\}$

Definition 2.3: Subgroup

A subset H of a group G is a subgroup if:

H is closed under the composition law of G, $\star$ , $x\star y\in H, \forall x,y\in H$ .
The identity element of G is in H, $e_{G}\in H$.
Every element of H has an inverse element of H, $x^{-1}\in H, \forall x\in H$ .

Definition 2.2: Cyclic

A group $(G,\star)$ is a Cyclic if there exists an element a of G such that $G=<a> =\{a^{n}: n \in \mathbb{Z}\}$ where $n \in \mathbb{Z}$ . We say that “a generates G” or that a is “a generator of G”.

If $(G,\star)$ is cyclic and $ord(a)$ is finite then $|G|=ord(a)$ , in otherwords the order of a cyclic group is the order of the element that generates it.

It is left as an exercise for the reader to check that $<r>=\{e,r,r^{2},r^{3}\}$ is a cyclic subgroup of $D_{8}$ . A great way to visualise <r> is by isolating a side of the rubiks cube.

Bibliography

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MTH2002 – Algebra – a course given at the University of Exeter by Prof Mohammed Saidi 2018