36 lines
808 B
Typst
36 lines
808 B
Typst
#import "@local/handout:0.1.0": *
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#show: handout.with(
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title: [Symmetric Groups],
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by: "Mark",
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)
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#include "parts/00 intro.typ"
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#pagebreak()
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#include "parts/01 cycle.typ"
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#pagebreak()
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#include "parts/02 groups.typ"
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#pagebreak()
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#include "parts/03 subgroup.typ"
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#pagebreak()
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= Bonus problems
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#problem()
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Show that $x in ZZ^+$ has a multiplicative inverse mod $n$ if and only if $gcd(x, n) = 1$
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#v(1fr)
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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 $S_n$. \
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Show that $tau sigma tau^(-1)$ = $(tau(sigma_1), tau(sigma_2), ..., tau(sigma_k))$ \
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#hint[$tau$ is a permutation, so $tau(x)$ is the value at position $x$ after applying $tau$.]
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#v(1fr)
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#problem()
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Show that the set ${ (1, 2), (1,2,...,n)}$ generates $S_n$.
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#v(1fr)
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