149 lines
3.5 KiB
Typst
Executable File
149 lines
3.5 KiB
Typst
Executable File
#import "@local/handout:0.1.0": *
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#import "@preview/cetz:0.4.2"
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#import "../macros.typ": *
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= Groups
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#definition()
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Before we continue, we must introduce a bit of notation:
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- $S_n$ is the set of permutations on $n$ objects.
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- $ZZ_n$ is the set of integers mod $n$.
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#problem()
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What are the elements of $S_3$? #hint[Use cycle notation] \
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How about $ZZ_8$?
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#v(1fr)
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#definition()
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A _group_ $(G, *)$ consists of a set $G$ and an operator $*$. \
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Groups always have the following properties:
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+ $G$ is closed under $*$. In other words, $a, b in G => a * b in G$.
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+ $*$ is _associative_: $(a * b) * c = a * (b * c)$ for all $a,b,c in G$
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+ There is an _identity_ $e in G$, so that $a * e = e * a = a$ for all $a in G$.
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+ For any $a in G$, there exists a $b in G$ so that $a * b = b * a = e$. $b$ is called the _inverse_ of $a$. \
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This element is written as $-a$ if our operator is addition and $a^(-1)$ otherwise.
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Any pair $(G, *)$ that satisfies these properties is a group.
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#problem()
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Is $(ZZ_5, +)$ a group? \
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Is $(ZZ_5, -)$ a group? \
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#note[$+$ and $-$ refer to the usual operations in modular arithmetic.]
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#v(1fr)
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#problem()
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What is the group with the fewest number of elements?
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#solution[
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Let $(G, star)$ be our group, where $G = {x}$ and $star$ is defined by $x star x = x$
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Verifying that the trivial group is a group is trivial.
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]
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#definition()
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$ZZ_n^times$ is the set of integers mod $n$ with multiplicative inverses. \
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We can prove that this is the set of integers smaller than $n$ and coprime to $n$. \
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For example, $ZZ_12^times = {1, 5, 7, 11}$.
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#problem()
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What are the elements of $ZZ^times_8$? \
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How about $ZZ^times_23$? #hint[23 is prime.]
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#v(1fr)
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#pagebreak()
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#problem()
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Show that function composition is associative
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#v(1fr)
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#problem()
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Show that $S_n$ is a group under composition.
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#v(1fr)
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#problem()
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Let $(G, *)$ be a group with finitely many elements, and let $a in G$. \
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Show that there is an $n$ in $in ZZ$ so that $a^n = e$ \
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#hint[
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$a^n = a * a * ... * a$ repeated $n$ times. \
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$a^(-n) = a^(-1) * a^(-1) * ... * a^(-1)$, where $a^(-1)$ is the inverse of $a$. \
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]
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#v(2mm)
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The smallest such $n$ defines the _order_ of $g$.
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#hint[
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We've already done a special case of this problem! \
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Find it in this handout, then rewrite your proof for an arbitrary (finite) group.
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]
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#v(1fr)
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#problem()
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What is the order of 5 in $(ZZ_25, +)$? \
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What is the order of 2 in $(ZZ_17^times, times)$? \
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#v(1fr)
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#pagebreak()
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#definition("Generator", label: "gendef")
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Let $G$ be a group, and let $g$ be an element of $G$. \
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We say $g$ is a _generator_ if every other element of $G$ may be written as a power of $g$. \
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#problem()
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Let $G$ be a group of $n$ elements. \
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If $g$ is a generator, what is its order? \
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Provide a proof.
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#v(1fr)
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#problem()
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Find the two generators in $(ZZ, +)$ \
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Then, find all generators of $(ZZ_5, +)$
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#v(1fr)
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#problem()
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How many groups have only one generator?
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#solution[
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Two: the trivial group and $(ZZ_2, +)$.
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]
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#v(1fr)
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#definition()
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Let $S$ be a subset of the elements in $G$. \
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We say that $S$ _generates_ $G$ if every element of $G$ may be written as a product of elements in $S$. \
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#note(type: "Note")[This is an extension of @gendef.]
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#problem()
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We've already found a few generating sets of $S_n$. What are they?
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#solution[
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The following sets generate $S_n$:
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- All transpositions
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- All transpositions of the form $(1, k)$
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- All adjacent transpositions
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#v(2mm)
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The smallest generating set of $S_n$ consists of the transposition $(12)$ and the $n$-cycle $(1,2,...,n)$. \
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The proof of this is a bonus problem later in the handout.
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]
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#v(1fr)
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