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@ -35,14 +35,14 @@
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\section{Bonus problems}
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\problem{}
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Show that $x$ has a multiplicative inverse mod $n$ iff $\text{gcd}(x, n) = 1$
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Show that $x \in \mathbb{Z}^+$ has a multiplicative inverse mod $n$ iff $\text{gcd}(x, n) = 1$
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\vfill
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\problem{}
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Let $\sigma = (\sigma_1 \sigma_2 ... \sigma_k)$ be a $k$-cycle in $S_n$, and let $\tau$ be an arbitrary element of $S_n$. \par
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Show that $\tau \sigma \tau^{-1}$ = $\bigl(\tau(\sigma_1), \tau(\sigma_2), ..., \tau(\sigma_k)\bigr)$ \par
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\hint{As usual, $\sigma$ is a permutation. Thus, $\sigma(x)$ is the value at position $x$ after applying $\sigma$.}
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\hint{As usual, $\tau$ is a permutation. Thus, $\tau(x)$ is the value at position $x$ after applying $\tau$.}
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\vfill
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@ -1,18 +1,7 @@
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\section{Introduction}
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\definition{Intuitive permutations}
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Intuitively, a \textit{permutation} is an ordered arrangement of a set of objects. \par
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For example, $123$, $312$, and $231$ are all permutations of 1, 2, and 3.
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\problem{}
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List all permutations on three objects. \par
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How many permutations of $n$ objects are there?
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\vfill
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\definition{Formal permutations}<permadef>
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\definition{}
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Let $\Omega$ be an arbitrary set of $n$ objects. \par
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A \textit{permutation} on $\Omega$ is a bijective map $f: \Omega \to \Omega$.
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@ -26,24 +15,31 @@ The permutation $[312]$ is given by a map $f$ defined by the following table:
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\item $f(3) = 2$
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\end{itemize}
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Similarly, the \textit{trivial permutation} $[123]$ is given by the identity map $f(x) = x$.
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\problem{}
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What map corresponds to the permutation $[321]$?
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List all permutations on three objects. \par
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How many permutations of $n$ objects are there?
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\vfill
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\problem{}
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What map corresponds to the permutation $[321]$?
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\vfill
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\problem{}
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Why do we define permutations as a \textit{bijective} map?
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What map corresponds to the \say{do-nothing} permutation? \par
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Write it as a function and in square-bracket notation. \par
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\note[Note]{We usually call this the \textit{trivial permutation}}
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\vfill
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\pagebreak
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We can visualize permutations with a diagram we'll call the \say{braid.}
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We can visualize permutations with a \textit{string diagram}, shown below. \par
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The arrows in this diagram denote the image of $f$ for each possible input.
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Two examples are below:
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@ -161,8 +157,8 @@ The rightmost diagram uses arbitrary, meaningless labels.
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It shouldn't be hard to see that despite the different \say{output} order (2134 and 1432), \par
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the same permutation is depicted in all three diagrams. This example demonstrates two things:
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\begin{itemize}[itemsep=2mm]
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\item First, the items of our set do not have any meaning. \par
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$\Omega$ is just a set of arbitrary \textit{things}, which we may label however we like.
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\item First, the names of the items in our set do not have any meaning. \par
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$\Omega$ is just a set of $n$ arbitrary things, which we may label however we like.
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\item Second, permutations are verbs. We do not care about the \say{output} of a certain permutation,
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we care about what it \textit{does}. We could, for example, describe the permutation above as
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@ -176,16 +172,10 @@ Why, then, do we order our elements when we talk about permutations? As noted be
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If we assign a natural order to the elements of $\Omega$ (say, 1234), we can identify permutations by simply listing
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their output:
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Clearly, $[1234]$ represents the trivial permutation, $[2134]$ represents \say{swap first two,}
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and $[4123]$ represents \say{cycle left.}
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and $[4123]$ represents \say{cycle right.}
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\problem{}
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Draw braids for $[4123]$ and $[2341]$.
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Draw string diagrams for $[4123]$ and $[2341]$.
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\vfill
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Finally, note that permutations (as defined in \ref{permadef}) are \textit{not} \say{orderings of a certain set.} \par
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They are defined as \textit{bijective maps}, which can be written as orderings of a given array. \par
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Remember: permutations are verbs!
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\pagebreak
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@ -2,8 +2,10 @@
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\section{Cycle Notation}
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\definition{Order}
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The \textit{order} of a permutation $f$ is the smallest $n$ so that $f^n(x) = x$ for all $x$. \par
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In other words, if we repeat this permutation $n$ times, we get back to where we started.
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The \textit{order} of a permutation $f$ is the smallest positive $n$ so that $f^n(x) = x$ for all $x$. \par
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In other words: if we repeat this permutation $n$ times, we get back to where we started. \par
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Note that the order is given by the \textit{smallest} positive integer $n$. There may be more than one!
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\vspace{2mm}
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@ -44,7 +46,7 @@ Naturally, the identity permutation has order one.
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\problem{}
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What is the order of $[2314]$? \par
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How about $[4321]$? \par
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\note[Note]{You shouldn't need to draw any braids to solve this problem.}
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\note[Note]{You shouldn't need to draw any strings to solve this problem.}
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\vfill
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@ -168,6 +170,9 @@ The permutation $[431265]$ is a bit more interesting---it contains of two cycles
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\end{center}
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Another name we'll often use for two-cycles is \textit{transposition}. \par
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Any permutation that swaps two adjacent elements is called a transposition. \par
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\problem{}
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Find all cycles in $[5342761]$.
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@ -417,7 +422,8 @@ Be careful.
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\problem{}
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Look at the last two permutations in \ref{insquare}, $(1234)$ and $(3412)$. \par
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These are \textit{identical}---they are the same cycle written in two different ways. \par
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List all other ways to write this cycle. \hint{There are two more.}
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List all other ways to write this cycle. \hint{There are two more.} \par
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\note{Also, note that the last two permutations in \ref{insquare} are the same.}
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\pagebreak
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@ -6,11 +6,11 @@ Before we continue, we must introduce a bit of notation:
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\item $S_n$ is the set of permutations on $n$ objects.
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\item $\mathbb{Z}_n$ is the set of integers mod $n$.
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\item $\mathbb{Z}_n^\times$ is the set of integers mod $n$ with multiplicative inverses, which is \par
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the set of integers smaller than $n$ and coprime to $n$\footnotemark{}\hspace{-1ex}. \par
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\item $\mathbb{Z}_n^\times$ is the set of integers mod $n$ with multiplicative inverses. \par
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In other words, it is the set of integers smaller than $n$ and coprime to $n$.\footnotemark{} \par
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For example, $\mathbb{Z}_{12}^\times = \{1, 5, 7, 11\}$.
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\footnotetext{We proved this in another handout, but you make take it as fact here.}
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\footnotetext{We proved this in another handout, but you may take it as fact here.}
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\end{itemize}
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\problem{}
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@ -26,7 +26,7 @@ Groups always have the following properties:
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\begin{enumerate}
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\item $G$ is closed under $\ast$. In other words, $a, b \in G \implies a \ast b \in G$.
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\item $\ast$ is associative: $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a,b,c \in G$
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\item $\ast$ is \textit{associative}: $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a,b,c \in G$
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\item There is an \textit{identity} $e \in G$, so that $a \ast e = a \ast e = a$ for all $a \in G$.
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\item For any $a \in G$, there exists a $b \in G$ so that $a \ast b = b \ast a = e$. $b$ is called the \textit{inverse} of $a$. \par
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This element is written as $-a$ if our operator is addition and $a^{-1}$ otherwise.
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@ -42,7 +42,7 @@ Is $(\mathbb{Z}_5, -)$ a group? \par
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\problem{}
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What is the smallest group?
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What is the group with the fewest elements?
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\begin{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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@ -63,7 +63,10 @@ What is the smallest group?
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\problem{}
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Show that function composition is associative
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\vfill
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\problem{}
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@ -82,7 +85,7 @@ The smallest such $n$ defines the \textit{order} of $g$.
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\begin{examplesolution}
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We've already done a special case of this problem! \par
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Look back through the handout and find it, then rewrite your proof for an arbitrary group.
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Find it in this handout, then rewrite your proof for an arbitrary (finite) group.
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\end{examplesolution}
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@ -116,34 +119,25 @@ We say $g$ is a \textit{generator} if every other element of $G$ may be written
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Say the size of a group $G$ is $n$. \par
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If $g$ is a generator, what is its order? \par
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Provide a proof.
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\vfill
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\problem{}
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Find the two generators in $(\mathbb{Z}, +)$ \par
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Then, find all generators of $(\mathbb{Z}_5, +)$
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\vfill
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\problem{}
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How many groups have only one generator?
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\begin{solution}
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The order of a generator must equal the order of its group.
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Only one: the trivial group. The inverse of a generator is also a generator!
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\end{solution}
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\vfill
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\problem{}
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Find the only generator of $(\mathbb{Z}^+, +)$ \par
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Then, find all generators of $(\mathbb{Z}_5, +)$
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\vfill
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\pagebreak
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\definition{}
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Let $S$ be a subset of the elements in $G$. \par
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@ -168,13 +162,4 @@ We've already found a few generating sets of $S_n$. What are they?
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\end{solution}
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\vfill
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\problem{}
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Find the smallest set that generates $(\mathbb{Z}^+, +)$. \par
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\vfill
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\problem{}
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Find the smallest set that generates $(\mathbb{Z}, +)$. \par
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\vfill
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\pagebreak
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@ -6,19 +6,18 @@ What elements do $S_2$ and $S_3$ share?
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Consider the sets $\{1, 2\}$ and $Omega_3 = \{1,2,3\}$. Clearly, $\{1, 2\} \subset \{1, 2, 3\}$. \par
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Consider the sets $\{1, 2\}$ and $\{1,2,3\}$. Clearly, $\{1, 2\} \subset \{1, 2, 3\}$. \par
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Can we say something similar about $S_2$ and $S_3$?
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\vspace{2mm}
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Looking at \ref{s2s3share}, we may want to say that $S_2 \subset S_3$ since every element of $S_2$ is in $S_3$. \par
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This reasoning, however, is not correct. Remember that $S_2$ and $S_3$ are \textit{groups}, not \textit{sets}: \par
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their elements come with structure.
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This however, isn't as interesting as it could be. Remember that $S_2$ and $S_3$ are \textit{groups}, not \textit{sets}: \par
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their elements come with structure, which the \say{subset} relation does not capture.
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\vspace{2mm}
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Therefore, the \say{subset} relation isn't particularly useful when applied to groups. \par
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We instead use a similar relation: subgroups.
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To account for this, we'll define a similar relation: subgroups.
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\definition{}
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Let $G$ and $G'$ be groups. We say $G'$ is a \textit{subgroup} of $G$ (and write $G' \subset G$) if the following are true:\par
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@ -73,19 +72,21 @@ Show that $S_3$ is a subgroup of $S_4$.
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\pagebreak
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\problem{}<firstindex>
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How many subgroups of $S_4$ are equal to $S_3$?
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\begin{solution}
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Four, since there are four ways to pick three things from $S_4$.
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\end{solution}
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How many subgroups of $S_4$ are behave like to $S_3$? \par
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\note{
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Of course, \say{behaves like} is a very hand-wavy relationship. \\
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Formally, this is called \textit{isomorphism}, but we'll formally define that
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in a later lesson.
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}
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\vfill
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\problem{}
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What is the order of $S_3$ and $S_4$? \par
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What are the orders of $S_3$ and $S_4$? \par
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How is this related to \ref{firstindex}?
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\begin{solution}
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@ -93,8 +94,8 @@ How is this related to \ref{firstindex}?
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\vspace{2mm}
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This solution is written using index notation, but the class
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doesn't yet need to know what it means.
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This solution is written using index notation, \par
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but the class doesn't need to know what it means yet.
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\end{solution}
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\vfill
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@ -108,16 +109,14 @@ How many instances of each does $S_4$ contain?
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\problem{}
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$(\mathbb{Z}_4, +)$ is also a subgroup of $S_4$. Find it! \par
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How many copies of $Z_4$ are in $S_4$? \par
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(You'll need to re-label elements, since we usually use different notation for $\mathbb{Z}_4$ and $S_4$).
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How many subgroups of $\mathbb{Z}_4$ are isomorphic to $S_4$?.
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\begin{solution}
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A good hint is \say{look at generators.}
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\vspace{4mm}
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There are four instances of $\mathbb{Z}_4$ in $S_4$, \par
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each of which is generated by a 4-cycle of $S_n$. \par
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There are four instances of $\mathbb{Z}_4$ in $S_4$, each of which is generated by a 4-cycle of $S_n$. \par
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(i.e, the group generated by $(1234)$ is isomorphic to $\mathbb{Z}_4$)
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\end{solution}
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