Rewrite symmetric groups

src/Advanced/Symmetric Groups/parts/02 groups.typ

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+#import "@local/handout:0.1.0": *
+#import "@preview/cetz:0.4.2"
+#import "../macros.typ": *
+
+= Groups (review)
+
+#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 $n$.
+
+- $ZZ_n^times$ is the set of integers mod $n$ with multiplicative inverses. \
+  In other words, it is the set of integers smaller than $n$ and coprime to $n$.#footnote[We proved this in another handout, but you may take it as fact here.] \
+  For example, $ZZ_12^times = {1, 5, 7, 11}$.
+
+#problem()
+What are the elements of $S_3$? #hint[Use cycle notation] \
+How about $ZZ_17^times$?
+
+#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.
+]
+
+#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.]
+
+#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[
+  Only one: the trivial group. The inverse of a generator is also a generator!
+]
+
+#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)