Algebra Qual Prep Week 3

Tags: #qual_algebra

Week 3: Sylow Theory

See the Presentation Schedule


  • 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?