173 lines
4.3 KiB
Typst
173 lines
4.3 KiB
Typst
#import "@local/handout:0.1.0": *
|
|
#import "@preview/cetz:0.4.2"
|
|
#import "../macros.typ": *
|
|
|
|
= Subgroups
|
|
|
|
#problem(label: "s2s3share")
|
|
What elements do $S_2$ and $S_3$ share?
|
|
|
|
#v(2cm)
|
|
|
|
|
|
Consider the sets $\{1, 2\}$ and $\{1,2,3\}$. Clearly, $\{1, 2\} subset \{1, 2, 3\}$. \
|
|
Can we say something similar about $S_2$ and $S_3$?
|
|
|
|
#v(2mm)
|
|
|
|
Looking at @s2s3share, we may want to say that $S_2 subset S_3$ since every element of $S_2$ is in $S_3$. \
|
|
This however, isn't as interesting as it could be. Remember that $S_2$ and $S_3$ are _groups_, not _sets_: \
|
|
their elements come with structure, which the "subset" relation does not capture.
|
|
|
|
#v(2mm)
|
|
|
|
To account for this, we'll define a similar relation: subgroups.
|
|
|
|
#definition("Subgroup")
|
|
Let $G$ and $G'$ be groups. We say $G'$ is a _subgroup_ of $G$ (and write $G' subset G$) if the following are true:\
|
|
(Note that $x, y$ are elements of $G$, and $x y$ is multiplication in $G$)
|
|
- the set of elements in $G'$ is a subset of the set of elements in $G$.
|
|
- the identity of $G$ is in $G'$
|
|
- $x,y in G' => x y in G'$
|
|
- $x in G' => x^(-1) in G'$
|
|
|
|
The above definition may look fairly scary, but the idea behind a subgroup is simple. \
|
|
Consider $S_3$ and $S_4$, the groups of permutations of $3$ and $4$ elements. \
|
|
|
|
#v(2mm)
|
|
|
|
Say we have a set of four elements and only look at the first three. \
|
|
$S_3$ fully describes all the ways we can arrange those three elements:
|
|
|
|
#table(
|
|
columns: (1fr,),
|
|
align: center,
|
|
stroke: none,
|
|
align(center, cetz.canvas({
|
|
import cetz.draw: *
|
|
|
|
let s = 0.7
|
|
|
|
set-style(content: (frame: "rect", stroke: none, fill: white, padding: .1))
|
|
content((0 * s, 0.5 * s), $1$, name: "1a")
|
|
content((1 * s, 0.5 * s), $2$, name: "2a")
|
|
content((2 * s, 0.5 * s), $3$, name: "3a")
|
|
content((3 * s, 0.5 * s), $4$, name: "4a")
|
|
|
|
content((0 * s, -2 * s), $2$, name: "2b")
|
|
content((1 * s, -2 * s), $3$, name: "3b")
|
|
content((2 * s, -2 * s), $1$, name: "1b")
|
|
content((3 * s, -2 * s), $4$, name: "4b")
|
|
|
|
|
|
// These arrows are wrong,
|
|
// but create a symmetric picture
|
|
markline(s, "1a", "1b")
|
|
markline(s, "2a", "3b")
|
|
markline(s, "3a", "2b")
|
|
markline(s, "4a", "4b", c: ogreen)
|
|
|
|
content(
|
|
(1 * s, -0.55 * s),
|
|
$S_3$,
|
|
fill: white,
|
|
stroke: oblue + 0.6mm,
|
|
padding: 1.3mm,
|
|
)
|
|
})),
|
|
)
|
|
|
|
|
|
#problem()
|
|
Show that $S_3$ is a subgroup of $S_4$.
|
|
|
|
#v(1fr)
|
|
#pagebreak()
|
|
|
|
|
|
|
|
#definition("Isomorphism")
|
|
Let $G$ and $H$ be groups. We say that $G$ and $H$ are _isomorphic_ (and write $G tilde.equiv H$) \
|
|
if there is a bijection $f: G -> H$ with the following properties:
|
|
- $f(e_G) = e_H$, where $e_G$ is the identity in $G$
|
|
- $f(x^(-1)) = f(x)^(-1)$ for all $x$ in $G$
|
|
- $f(x y) = f(x) f(y)$ for all $x, y$ in $G$
|
|
|
|
Intuitively, you can think of isomorphism as a form of equivalence. \
|
|
If two groups are isomorphic, they only differ by the names of their elements. \
|
|
The function $f$ above tells us how to map one set of labels to the other.
|
|
|
|
|
|
|
|
#problem()
|
|
Show that $ZZ_7^times$ and $ZZ_9^times$ are isomorphic.
|
|
#hint[
|
|
Build a bijection with the above properties. \
|
|
Remember that a group is fully defined by its multiplication table.
|
|
]
|
|
#v(1fr)
|
|
|
|
|
|
#problem()
|
|
Show that $ZZ_10^times$, $ZZ_5^times$, and $ZZ_4$ are isomorphic.
|
|
#hint[
|
|
Build a bijection with the above properties. \
|
|
Remember that a group is fully defined by its multiplication table.
|
|
]
|
|
|
|
|
|
#v(1fr)
|
|
|
|
#problem()
|
|
Show that isomorphism is transitive. \
|
|
That is, if $A tilde.equiv B$ and $B tilde.equiv C$, then $A tilde.equiv C$.
|
|
|
|
#v(1fr)
|
|
#pagebreak()
|
|
|
|
|
|
#problem(label: "firstindex")
|
|
How many subgroups of $S_4$ are isomorphic to $S_3$? \
|
|
|
|
#v(1fr)
|
|
|
|
|
|
|
|
|
|
#problem()
|
|
What are the orders of $S_3$ and $S_4$? \
|
|
How is this related to @firstindex?
|
|
|
|
#solution[
|
|
$|S_4| = |S_3| times [S_4 : S_3]$
|
|
|
|
#v(2mm)
|
|
|
|
This solution is written using index notation, \
|
|
but the class doesn't need to know what it means yet.
|
|
]
|
|
|
|
#v(1fr)
|
|
|
|
#problem()
|
|
$S_4$ also has $S_2$ and the trivial group as subgroups. \
|
|
How many instances of each does $S_4$ contain?
|
|
|
|
#v(1fr)
|
|
|
|
|
|
#problem()
|
|
$(ZZ_4, +)$ is also a subgroup of $S_4$. Find it! \
|
|
How many subgroups of $ZZ_4$ are isomorphic to $S_4$?
|
|
|
|
#solution[
|
|
A good hint is "look at generators."
|
|
|
|
#v(4mm)
|
|
|
|
There are four instances of $ZZ_4$ in $S_4$, each of which is generated by a 4-cycle of $S_n$. \
|
|
(i.e, the group generated by $(1234)$ is isomorphic to $ZZ_4$)
|
|
]
|
|
|
|
#v(1fr)
|