From 30713a7916110932b49fe89c978d37dff65678c4 Mon Sep 17 00:00:00 2001 From: Mark Date: Tue, 30 Sep 2025 18:54:04 -0700 Subject: [PATCH] Symmetric edits --- src/Advanced/Symmetric Groups/main.typ | 2 +- .../Symmetric Groups/parts/00 intro.typ | 2 +- .../Symmetric Groups/parts/01 cycle.typ | 8 +++--- .../Symmetric Groups/parts/02 groups.typ | 27 ++++++++++++------- 4 files changed, 24 insertions(+), 15 deletions(-) diff --git a/src/Advanced/Symmetric Groups/main.typ b/src/Advanced/Symmetric Groups/main.typ index 0d5db50..b8e795d 100644 --- a/src/Advanced/Symmetric Groups/main.typ +++ b/src/Advanced/Symmetric Groups/main.typ @@ -20,7 +20,7 @@ = Bonus problems #problem() -Show that $x in ZZ^+$ has a multiplicative inverse mod $n$ iff $gcd(x, n) = 1$ +Show that $x in ZZ^+$ has a multiplicative inverse mod $n$ if and only if $gcd(x, n) = 1$ #v(1fr) diff --git a/src/Advanced/Symmetric Groups/parts/00 intro.typ b/src/Advanced/Symmetric Groups/parts/00 intro.typ index 9ab3ffc..10ba583 100644 --- a/src/Advanced/Symmetric Groups/parts/00 intro.typ +++ b/src/Advanced/Symmetric Groups/parts/00 intro.typ @@ -45,7 +45,7 @@ How many permutations of $n$ objects are there? #v(1fr) #problem() -What map corresponds to the permutation that produces the array `312` from the array `123`? +What map corresponds to the permutation that produces the array `231` from the array `123`? #v(1fr) diff --git a/src/Advanced/Symmetric Groups/parts/01 cycle.typ b/src/Advanced/Symmetric Groups/parts/01 cycle.typ index bed8fd4..4520f6d 100755 --- a/src/Advanced/Symmetric Groups/parts/01 cycle.typ +++ b/src/Advanced/Symmetric Groups/parts/01 cycle.typ @@ -100,7 +100,7 @@ How about $[321][213][231]$? \ Rewrite these compositions as one permutation in square brackets. #solution([ - - $[1324][4321]$ is $[4321]$ + - $[1324][4321]$ is $[4231]$ - $[321][213][231]$ is $[123]$ ]) @@ -501,7 +501,7 @@ List all other ways to write this cycle. \ #definition("Inverse") -The _inverse_ of a permitation $f$ is a permutation $g$ that "un-does" $f$. \ +The _inverse_ of a permutation $f$ is a permutation $g$ that "un-does" $f$. \ This means that $g(f(x)) = x$ for all $x$. #problem() @@ -550,7 +550,7 @@ Show that any cycle $(123...n)$ is equal to the product $(12)(23)...(n-1, n)$. Write $(7126453)$ as a product of transpositions. \ #solution[ - Move elements one at a time, and using the last position as temporary storage. + Move elements one at a time, using the last position as temporary storage. We get $(71)(72)(76)(74)(75)(73)$. Other solutions are possible. \ @@ -589,7 +589,7 @@ Show that any permutation is a product of transpositions of the form $(1, k)$. \ #problem(label: "oneplustrans") -Show that any transposition $(a, b)$ is equal to the product $(a, a+1)(a+1, b)(a, a+1)$. +Show that any transposition $(a, b)$ is equal to the product $(a, a+1)(a+1, b)(a, a+1)$ whenever $a + 1 != b$. #solution[ This is the same as @onetrans, diff --git a/src/Advanced/Symmetric Groups/parts/02 groups.typ b/src/Advanced/Symmetric Groups/parts/02 groups.typ index be7ad56..85e96ea 100755 --- a/src/Advanced/Symmetric Groups/parts/02 groups.typ +++ b/src/Advanced/Symmetric Groups/parts/02 groups.typ @@ -2,20 +2,16 @@ #import "@preview/cetz:0.4.2" #import "../macros.typ": * -= Groups (review) += Groups #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 $ZZ_17^times$? +How about $ZZ_8$? #v(1fr) @@ -47,6 +43,16 @@ What is the group with the fewest number of elements? Verifying that the trivial group is a group is trivial. ] + +#definition() +$ZZ_n^times$ is the set of integers mod $n$ with multiplicative inverses. \ +We can prove that this is the set of integers smaller than $n$ and coprime to $n$. \ +For example, $ZZ_12^times = {1, 5, 7, 11}$. + +#problem() +What are the elements of $ZZ^times_8$? \ +How about $ZZ^times_23$? #hint[23 is prime.] + #v(1fr) #pagebreak() @@ -66,8 +72,11 @@ Show that $S_n$ is a group under composition. #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.] +Show that there is an $n$ in $in ZZ$ so that $a^n = e$ \ +#hint[ + $a^n = a * a * ... * a$ repeated $n$ times. \ + $a^(-n) = a^(-1) * a^(-1) * ... * a^(-1)$, where $a^(-1)$ is the inverse of $a$. \ +] #v(2mm) @@ -111,7 +120,7 @@ Then, find all generators of $(ZZ_5, +)$ How many groups have only one generator? #solution[ - Only one: the trivial group. The inverse of a generator is also a generator! + Two: the trivial group and $(ZZ_2, +)$. ] #v(1fr)