# Algebra Qual Prep Week 3

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 7-subgroup and a central Sylow 11-subgroup.
• 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?