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Section Day 12
This is an outline of the topics we covered in the twelfth 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 6/25
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.
Use group actions to prove theorems about groups
Apply the Sylow Theorems to determine properties of or classify groups of a given order
Algebraist of the Day.
Ludwig Sylow , 1832-1918
Norwegian mathematician and high school teacher
Substituted at Christiana University in 1862, where he had a impact on Lie’s early understanding of group theory
Eventually gained a university position at Christiana University at age 65
Theorem 128 . Sylow Theorem I.
A finite group
\(G\) has a
\(p\) -Sylow subgroup for every prime
\(p\text{,}\) each
\(p\) -subgroup of
\(G\) is contained in a
\(p\) -Sylow subgroup, and
\(G\) has a
\(p\) -subgroup of every order
\(p^k\) dividing
\(|G|\text{.}\)
Theorem 129 . Sylow Theorem II.
For each prime
\(p\text{,}\) the
\(p\) -Sylow subgroups of a finite group
\(G\) are all conjugate to each other.
Theorem 130 . Sylow Theorem III.
For each prime \(p\text{,}\) let \(n_p\) be the number of \(p\) -Sylow subgroups of a finite group \(G\text{.}\) Write \(|G| = p^k m\) where \(p\) does not divide \(m\text{.}\) Then
\begin{equation*}
n_p \mid m \quad \text{and}\quad n_p \equiv 1 \bmod p\text{.}
\end{equation*}
Overview of the Theorems and Proof Plan.
Sylow I in groups of order 12: We saw this in the Daily Prep.
Sylow II in groups of order 12:\(\)
Sylow III in groups of order 12:\(\)
Groups actions for the proof: let
\(P,Q\) be
\(p\) -Sylow subgroups of
\(G\text{.}\)
Sylow I
\(p\) -subgroup \(H\)
\(G/H\) left mult.
Sylow II
\(p\) -Sylow subgroup \(Q\)
\(G/P\) left mult.
Sylow III (\(n_p\mid m\) )
\(G\) \(\Syl_p(G)\) conjugation
Sylow III (\(n_p\equiv 1 \pmod p\) )
\(P\) \(\Syl_p(G)\) conjugation
Some consequences of the Sylow Theorems:
Example 131 . No Group of Order 30 is Simple.
Example 132 . Groups of Order 12 Have a Normal Sylow Subgroup.
Example 133 . Groups of Order 24 Have a Normal Subgroup of Size 4 or 8.
