Groups

Sets

Before we can dive into groups, we first must know a few things. The first is what a set is.
A set is a list of items, it can be ordered or unordered. There are a few ways to notate a set, one includes curly braces. Anything in the set goes in the curly braces separated by commas, these are called the elements of the set.

For example, $S = {1, 2, 3, 4, 5}$ . A set is usually denoted by a capital letter. A set can hold anything, another example of a set is $F$ where $F = {◯, ◻, △}$ . A set can even hold other sets, $E = {{a, b, c}, {d, e, f}}$ .

Set Operations

Sets have operations and relations on them just like numbers. Similar to how $4 < 5$ or $4 + 3 > 1$ , sets have relations like $Z \subset R$ , ${1, 2, 3, 4} \cup {0, 4, 5, 6} = {0, 1, 2, 3, 4, 5, 6}$ . First we will go over the operations and relations then some famous sets.

The symbol $\subseteq$ is the subset symbol. A set $S$ is a subset of another set, $X$ , if $X$ contains every element in $S$ . If $X$ contains some elements not in $S$ as well, then $S$ is the proper subset of $X$ , $S \subset X$ .

The symbol $\supseteq$ is the exact opposite of subset. It is the superset symbol. A superset $X$ of a set $S$ contains all the elements of $S$ and possibly more, $X \supseteq S$ .

$\cup$ is the union symbol. It is an operator that combines sets like a venn diagram. The union of two sets, $X$ and $S$ , is notated as $X \cup S$ and is a new set that contains the elements of both sets with no repeats. If each set is one circle of the venn diagram then the union of the sets is the set containing both of the circles of the venn diagram.

$\cap$ is the intersection symbol. It is an operator that cuts sets like a venn diagram. The intersection of two sets, $X$ and $S$ , is notated as $X \cap S$ and is a new set that contains only elements that both $X$ and $S$ have in common. In the venn diagram example, the intersection is the part in the middle that touches both circles.

Finally, $\in$ is the set inclusion operator. It is a binary operator with an element on the left and a set on the right. It denotes when an object is an element of a set. For example, $1 \in {1, 2, 3}$ but $1 \notin {0, 2, 4, 6, 8, . . .}$ .

Famous Numerical Sets

$N$ is the set of all natural numbers ( $1$ , $2$ , $3$ , $4$ , ...); it is debated whether or not $N$ includes 0.
$Z$ is the set of all integers (..., $- 3$ , $- 2$ , $- 1$ , $0$ , $1$ , $2$ , $3$ , ...).
$Q$ is the set of all rational numbers ( $\frac{1}{2}$ , $164$ , $- 18.965$ , and more), numbers of the form $\frac{a}{b}$ when $a$ and $b$ are both integers and $b \neq 0$ .
$R$ is the set of all real numbers ( $e$ , $- 54.154$ , $π$ , $γ$ ).
$C$ is the set of all complex numbers, ( $4 - 3 i$ , $12.554 i$ , $e - π i$ ).
$N \subset Z \subset Q \subset R \subset C$

But what are groups?

Groups are a combination of a set and an operation (notated as an ordered pair) that satisfy 4 axioms. Let's consider the abstract group and group operation, $(G, \circ)$ . Note that $G$ can be any group and the $\circ$ operation is any operation that goes with $G$ . It is similar to a variable in that we have not defined it.

Closure: for any elements $a, b \in G$ , $a \circ b \in G$ . Using the group operation on any two elements in the group leads to a new element that is also in the group.
Associativity: for any three elements, $a, b, c \in G$ , $(a \circ b) \circ c = a \circ (b \circ c)$ . The group operation is associative.
Identity: there exists one and only one element $e \in G$ called the "identity" that satisfies the following property: for any element $a \in G$ , $a \circ e = a$ .
Inverse: for any element $a \in G$ , there exists one and only one other element $a^{- 1} \in G$ such that $a \circ a^{- 1} = e$ .

Let's look at an example.

Consider $(Z, +)$ . Is the set of all integers a group with operation addition? Let's find out.

Well for the first axiom, any integer plus any other integer is an integer so that works.
For the second axiom, we can pick three integers, for example, $a = 2, b = - 5, c = 10$ now we compute $(a + b) + c$ and $a + (b + c)$ . $(a + b) + c = (2 - 5) + 10 = - 3 + 10 = 7$ , and $a + (b + c) = 2 + (- 5 + 10) = 2 + (5) = 7$ . We now know that they are the same, therefore addition among the integers is associative.
The third axiom is the identity element. In the case of the integers and addition, the identity is $0$ . Any integer plus $0$ is the integer itself.
The final axiom is the inverse element. Consider the integer $a$ with inverse $a^{'}$ , we know that $a + a^{'} = 0$ , therefore $a^{'} = - a$ . So the inverse of any integer is its opposite.

The integers are a group over addition.

Significance

There are many groups, some of which do not have numbers. For example, the group of symmetries of an equilateral triangle is a group. Before we look at a more fun, real world example, let's prove one thing.

For a group, $(G, \circ)$ , and elements, $a$ and $b$ , the equation $a \circ x = b$ has a solution in $G$ .
But how do we prove it?

$a \circ x = b \Rightarrow a^{- 1} \circ (a \circ x) = a^{- 1} \circ (b)$ and by axiom 2, we can rewrite this as $(a^{- 1} \circ a) \circ x = a^{- 1} \circ b$ and by axiom 4, we can rewrite this as $e \circ x = a^{- 1} \circ b$ (where $e$ is the identity element) and by axiom 3, we can rewrite this as $x = a^{- 1} \circ b$ .

Now this is only one example of many cool examples of the power of groups, but consider a Rubik's Cube.

CC-BY-SA, Booyabazooka, Wikipedia, 30 November 2006, "Rubik's Cube".

A Rubik's Cube is a group with the operation of composition. The elements of the Rubik's Cube group are all the possible states of the cube. That means that the group has approximately 43 quintillion elements. This is a special type of group as it can be generated with only 6 of the states. That is, $⟨ L, R, F, B, D, U ⟩$ is called the generating set of the Rubik's Cube group as any element of the group can be made from a combination of the elements in the generating set.

But what is the operation? I mean I said that it is a composition but then I said that it is a combination, so what is it?
Well usually the operation is omitted from equations, for example, the state made by moving the front face clockwise then the right face clockwise is notated as $F R$ . The identity element is the solved state of the cube. Let's say that we have two elements from the group (any two states of the cube), $Ω$ and $Θ$ , the statement $Ω Θ$ is the same as the element that is made by doing the moves to get from the solved state to $Ω$ and then doing the moves that it takes to get from the solved state to $Θ$ .

Using our proof above and applying it here, we can say that for $Ω x = Θ$ , $x$ is the move that it takes to get from $Ω$ to $Θ$ . Using the solution, $x$ is $Ω^{'} Θ$ .

We can see that anything we prove for groups in general will work for every group. We have essentially abstracted away many of the specifics of many parts of math and allowed ourselves to think abstractly and do "multitask" with math.
Group theory has lots of power in higher math and the world itself.