Tags: #qual_algebra
Week 3: Sylow Theory
Topics
- The 3 Sylow theorems
- Showing groups are abelian
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Classification:
- FTFGAG
- Recognizing direct and semidirect products
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Groups of specified orders:
- \(p\)
- \(pq\)
- \(pqr\)
- \(p^2\)
- \(p^2 q\)
- Sketch a proof for each of the Sylow theorems.
Unsorted Questions
- Identify \(\mathop{\mathrm{Aut}}_{\mathsf{Grp}}(\bigoplus_{i=1}^n {\mathbb{Z}}/p)\) 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.
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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?