Tags: #qual_algebra
Week 3: Sylow Theory
Topics
 The 3 Sylow theorems
 Showing groups are abelian

Classification:
 FTFGAG
 Recognizing direct and semidirect products

Groups of specified orders:
 \(p\)
 \(pq\)
 \(pqr\)
 \(p^2\)
 \(p^2 q\)
 Sketch a proof for each of the Sylow theorems.
Unsorted Questions
 Identify \(\mathop{\mathrm{Aut}}_{\mathsf{Grp}}(\bigoplus_{i=1}^n {\mathbf{Z}}/p)\) as a matrix group and determine its size.
 State the 3 Sylow theorems
 Show that a group of order \(pq\) where \(p > q\). Show that \(G\) has a nontrivial proper normal subgroup,
 Show that any finite abelian group is isomorphic to the direct product of its Sylow subgroups
 . Assume that \(G\) is a group of order \(231 = 3\times 7\times 11\). Show that \(G\) contains a normal Sylow 7subgroup and a central Sylow 11subgroup.

If \(L/k\) is an abelian Galois extension of degree \(540 = 2^2 \times 3^3\times 5\), what are the possible Galois groups \({ \mathsf{Gal}}(L/k)\)?
 Are there any intermediate fields \(E\) for which \(L/E\) is a quadratic extension?