Applications of the Sylow Theorems- Groups of Order 12 (Algebra 1: Lecture 16 Video 1)

Lecture 16: In this lecture we gave several applications of the Sylow Theorems.  We started by giving some applications to groups of order 12.  We then gave applications to groups of order p^2*q and then we gave some applications to groups of order 24.  In the last video we gave two applications to matrix groups.  We gave one application to Sylow p-subgroups of GL(2,p), and then one application showing that the Heisenberg group H(Z/pZ) is the unique Sylow p-subgroup of the group of invertible upper-triangular 3 x 3 matrices over Z/pZ.
Reading: One of the results we proved about groups of order 12 is an example on page 144 of Dummit and Foote.  The application we gave to groups of order p^2*q is also on that page.  We also proved Theorems 5.4 and 5.5 of Conrad's notes on 'Applications of Sylow Theorems'.  The examples we gave related to groups of matrices are Theorems 2.4 and 2.7 of Conrad's notes.
At this point, you should read all of Section 4.5 of Dummit and Foote.  I would also highly encourage you to read over all of Conrad's 'Applications of Sylow Theorems' notes.  There may be some details you do not understand because they are related to things we have not covered (for example about the group Aff(Z/(p)) and about semidirect products), so if there is something unfamiliar, feel free to skip over it for now.