#import "@local/handout:0.1.0": *
#import "@preview/cetz:0.4.2"
#import "../macros.typ": *

= Groups

#definition()
Before we continue, we must introduce a bit of notation:
- $S_n$ is the set of permutations on $n$ objects.
- $ZZ_n$ is the set of integers mod $n$.

#problem()
What are the elements of $S_3$? #hint[Use cycle notation] \
How about $ZZ_8$?

#v(1fr)

#definition()
A _group_ $(G, *)$ consists of a set $G$ and an operator $*$. \
Groups always have the following properties:

+ $G$ is closed under $*$. In other words, $a, b in G => a * b in G$.
+ $*$ is _associative_: $(a * b) * c = a * (b * c)$ for all $a,b,c in G$
+ There is an _identity_ $e in G$, so that $a * e = e * a = a$ for all $a in G$.
+ For any $a in G$, there exists a $b in G$ so that $a * b = b * a = e$. $b$ is called the _inverse_ of $a$. \
  This element is written as $-a$ if our operator is addition and $a^(-1)$ otherwise.

Any pair $(G, *)$ that satisfies these properties is a group.

#problem()
Is $(ZZ_5, +)$ a group? \
Is $(ZZ_5, -)$ a group? \
#note[$+$ and $-$ refer to the usual operations in modular arithmetic.]

#v(1fr)

#problem()
What is the group with the fewest number of elements?

#solution[
  Let $(G, star)$ be our group, where $G = {x}$ and $star$ is defined by $x star x = x$

  Verifying that the trivial group is a group is trivial.
]


#definition()
$ZZ_n^times$ is the set of integers mod $n$ with multiplicative inverses. \
We can prove that this is the set of integers smaller than $n$ and coprime to $n$. \
For example, $ZZ_12^times = {1, 5, 7, 11}$.

#problem()
What are the elements of $ZZ^times_8$? \
How about $ZZ^times_23$? #hint[23 is prime.]

#v(1fr)
#pagebreak()





#problem()
Show that function composition is associative

#v(1fr)

#problem()
Show that $S_n$ is a group under composition.

#v(1fr)

#problem()
Let $(G, *)$ be a group with finitely many elements, and let $a in G$. \
Show that there is an $n$ in $in ZZ$ so that $a^n = e$ \
#hint[
  $a^n = a * a * ... * a$ repeated $n$ times. \
  $a^(-n) = a^(-1) * a^(-1) * ... * a^(-1)$, where $a^(-1)$ is the inverse of $a$. \
]

#v(2mm)

The smallest such $n$ defines the _order_ of $g$.

#hint[
  We've already done a special case of this problem! \
  Find it in this handout, then rewrite your proof for an arbitrary (finite) group.
]

#v(1fr)

#problem()
What is the order of 5 in $(ZZ_25, +)$? \
What is the order of 2 in $(ZZ_17^times, times)$? \

#v(1fr)
#pagebreak()




#definition("Generator", label: "gendef")
Let $G$ be a group, and let $g$ be an element of $G$. \
We say $g$ is a _generator_ if every other element of $G$ may be written as a power of $g$. \

#problem()
Let $G$ be a group of $n$ elements. \
If $g$ is a generator, what is its order? \
Provide a proof.

#v(1fr)

#problem()
Find the two generators in $(ZZ, +)$ \
Then, find all generators of $(ZZ_5, +)$

#v(1fr)

#problem()
How many groups have only one generator?

#solution[
  Two: the trivial group and $(ZZ_2, +)$.
]

#v(1fr)

#definition()
Let $S$ be a subset of the elements in $G$. \
We say that $S$ _generates_ $G$ if every element of $G$ may be written as a product of elements in $S$. \
#note(type: "Note")[This is an extension of @gendef.]

#problem()
We've already found a few generating sets of $S_n$. What are they?

#solution[
  The following sets generate $S_n$:
  - All transpositions
  - All transpositions of the form $(1, k)$
  - All adjacent transpositions

  #v(2mm)

  The smallest generating set of $S_n$ consists of the transposition $(12)$ and the $n$-cycle $(1,2,...,n)$. \
  The proof of this is a bonus problem later in the handout.
]

#v(1fr)