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Section Daily Prep 12
We will use group actions to prove the Sylow theorems, which are some of the most powerful tools in finite group theory for understanding the structure of finite groups. In this activity you’ll practice more with group actions to get ready for the results in class.
Objectives: Basic Learning Objectives
Before our class meeting, you should use the resources below to be able to learn the following. You should be reasonably fluent with these; we’ll answer some questions on them in class but not reteach them in detail.
State and instantiate the definition of Sylow
\(p\) -subgroup and Sylow subgroup.
Identify the Sylow
\(p\) -subgroups of a given finite group and prime
\(p\text{.}\)
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Gallian, Chapter 24, pp. 409-411
Review Day 11 notes as needed
Figure 126. Reference Video for \(p\) -Group Basics
Subsection Important Terms
Definition 127 . Sylow Subgroups.
Sylow Subgroups
Let
\(G\) be a finite group and
\(p\) a prime. A
Sylow \(p\) -subgroup of
\(G\) is a subgroup of
\(G\) of order
\(p^n\text{,}\) where
\(p^n\) is the largest power of
\(p\) that divides the order of
\(G\text{.}\) A
Sylow subgroup of
\(G\) is a Sylow
\(p\) -subgroup for some prime
\(p\text{.}\) The set of all Sylow
\(p\) -subgroups of
\(G\) is denoted
\(\Syl_p(G)\text{.}\)
