Galois Theory HT25, Groups

Created: November 10, 2024 | Updated: April 17, 2025 | About these notes | View in context | Study these flashcards

Flashcards

Signs of permutations

@Define the sign of a permutation $\sigma \in S _ n$.
\[\begin{aligned} \text{sgn} &: S_n \to \pm 1 \\\\ \sigma &\mapsto \begin{cases} 1 &\text{if } \sigma \in A_n \\\\ -1 &\text{if } \sigma \notin A_n \end{cases} \end{aligned}\]

Properties of cyclic groups

@State a useful characterisation of cyclic groups which is useful for:

  • Determining the number of intermediate subfields when the Galois group of a field extension is cyclic
  • Showing groups (e.g. the multiplicative group of a finite field) is cyclic

A finite abelian group $G$ is cyclic iff there is one subgroup for each divisor of $ \vert G \vert $.

@State the automorphisms of $\mathbb Z / n \mathbb Z$.
\[\text{Aut}(\mathbb Z / n \mathbb Z) \cong (\mathbb Z / n \mathbb Z)^\times\]

Transitive subgroups of $S _ n$

What are the transitive subgroups of $S _ 3$?
  • $S _ 3$ (order 6)
  • $C _ 3$ (order 3)

What are the transitive subgroups of $S _ 4$?
  • $S _ 4$ (order 24)
  • $A _ 4$ (order 12)
  • $D _ 8$ (order 8)
  • $V _ 4$ (order 4)
  • $C _ 4$ (order 4)

What are the transitive subgroups of $S _ 5$?
  • $S _ 5$ (order 120)
  • $A _ 5$ (order 60)
  • $F _ {20} = \mathbb Z/5\mathbb Z \rtimes (\mathbb Z/5\mathbb Z)^\times$ (order 20)
  • $D _ {10}$ (order 10)
  • $C _ 5$ (order 5)


Related posts