Symmetric edits

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)