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Section Day 6
This is an outline of the topics we covered in the sixth day of class. The skeleton notes are in a handout, which can be printed out using the printer icon at the top right of its section of the page for filling in during class. Filled notes for each day will be posted after class to Canvas.
Handout Thursday 6/4
Objectives: Advanced Learning Outcomes
During our class meeting, we will work on learning the following. Fluency with these is not expected or required before class.
Given a subgroup
\(H\) of a group
\(G\text{,}\) compute the left (or right) cosets of
\(H\) in
\(G\text{.}\)
Utilize properties of cosets in computation and proof
State and instantiate the definition of: index of a subgroup
\(H\) in
\(G\)
State and apply Lagrange’s Theorem
Reminders/Announcements.
Weekly Practice 3 due Tuesday
Video Project Group Selection due Tuesday
Proposition 60 . Boring But Useful Coset Properties.
Let \(G\) be a group, \(H\leq G\text{,}\) and \(a,b\in G\text{.}\) Then
Cosets contain their representative:
A coset is H iff its representative is in H:
Cosets are equal iff each contains the other’s representative:
Cosets are equal or disjoint:
Another useful property for checking coset equality:
Cosets have the same size:
Left and right cosets are equal iff H is fixed by conjugation by the representative:
Cosets are usually not subgroups:
Definition 61 .
Let
\(G\) be a group and
\(H\leq G\text{.}\) The index of
\(H\) in
\(G\text{,}\) denoted
\([G:H]\text{,}\) is the number of distinct left (or right) cosets of
\(H\) in
\(G\text{.}\)
Theorem 62 . Lagrange’s Theorem.
Let \(G\) be a finite group and \(H\leq G\text{.}\) Then \(|H|\) divides \(|G|\) and the index of \(H\) in \(G\) is
\begin{equation*}
[G:H]=|G|/|H|\text{.}
\end{equation*}
Proof.
Corollary 63 . Tower Law.
Suppose
\(K\leq H\leq G\text{.}\) Then
\([G:K]=[G:H][H:K]\text{.}\)
