#import "@local/handout:0.1.0": *

#show: handout.with(
  title: [Symmetric Groups],
  by: "Mark",
)

#include "parts/00 intro.typ"
#pagebreak()

#include "parts/01 cycle.typ"
#pagebreak()

#include "parts/02 groups.typ"
#pagebreak()

#include "parts/03 subgroup.typ"
#pagebreak()

= Bonus problems

#problem()
Show that $x in ZZ^+$ has a multiplicative inverse mod $n$ if and only if $gcd(x, n) = 1$

#v(1fr)

#problem()
Let $sigma = (sigma_1 sigma_2 ... sigma_k)$ be a $k$-cycle in $S_n$, and let $tau$ be an arbitrary element of $S_n$. \
Show that $tau sigma tau^(-1)$ = $(tau(sigma_1), tau(sigma_2), ..., tau(sigma_k))$ \
#hint[$tau$ is a permutation, so $tau(x)$ is the value at position $x$ after applying $tau$.]

#v(1fr)
#problem()
Show that the set ${ (1, 2), (1,2,...,n)}$ generates $S_n$.
#v(1fr)