Galois Theory HT25, Background from group theory

Created: November 10, 2024 | Updated: May 21, 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:

  1. Determining the number of intermediate subfields when the Galois group of a field extension is cyclic
  2. Showing groups (e.g. the multiplicative group of a finite field) is cyclic
  3. Showing a group is not cyclic

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

  1. Count the number of divisors, each of which corresponds to a subgroup / subfield
  2. Show that there is exactly one subgroup for each divisor
  3. Show that there are two distinct subgroups of the same order

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

Suppose $g^a = g^b$ where $g$ is an element of a cyclic group. What can you conclude about $a$ and $b$?

They are equivalent modulo $ \vert G \vert $.

Transitive subgroups of $S _ n$

@Define what it means for a finite subgroup $G$ of $S _ n$ to be transitive.

For any $x \in X$, $G \cdot x = X$.

Equivalently, for any $x$ and $y$, there is some $g$ such that $g(x) = y$.

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)

Group actions

Suppose $G$ is a group that acts on a set $X$. @Define what it means for $Y \subseteq X$ to be $G$-stable.

For all $y \in Y$, $g \cdot y \in Y$.

(Hence also $G$ acts on $Y$ by restriction).

Suppose:

  • $K/F$ is a field extension
  • $G = \text{Gal}(K/F)$

In what way does $G$ act on $K$?

$G$ is the set of all $F$-linear automorphisms of $K$, so it acts on $K$ via

\[g \cdot k = g(k)\]

Suppose:

  • $K/F$ is a field extension
  • $G = \text{Gal}(K/F)$
  • $f \in F[t]$
  • $V(f) := \{\alpha \in K \mid f(\alpha) = 0\}$

@Prove that $V(f)$ is a $G$-stable subset of $K$ and therefore that $G$ also acts on $V(f)$.

Let $\sigma \in G$ and $\alpha \in V(f)$. Suppose

\[f = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\]

Then:

\[\begin{aligned} f(\sigma(\alpha)) &= \sum^n_{i = 0} a_i \sigma(\alpha)^i \\\\ &= \sigma \left(\sum^n_{i = 1} a_i \alpha^i\right) &&(\star)\\\\ &= \sigma(f(\alpha)) \\\\ &= \sigma(0) \\\\ &= 0 \end{aligned}\]

where $(\star)$ is justified by the fact that $\sigma$ is an $F$-linear ring homomorphism. Hence $\sigma(\alpha) \in V(F)$ for all $\alpha \in V(f)$, so $V(f)$ is $G$-stable and hence $G$ also acts on $V(f)$.

Suppose:

  • $G$ is a group
  • $X$ is a set
  • $x \in X$

@Define the orbit map $\pi _ x$ and connect it to $\text{Stab} _ G(x)$.
\[\begin{aligned} \pi_x &: G \to G \cdot x \\\\ g &\mapsto g\cdot x \end{aligned}\]

Then

\[\text{Stab}_G(x) = \pi_x^{-1}(x)\]

Suppose:

  • $G$ is a group
  • $X$ is a set
  • $x \in X$
  • $g \in G$
  • $\pi _ x$ is the orbit map given by $\pi _ x(g) = g \cdot x$

@Prove that:

\[\pi_x^{-1}(g \cdot x) = g \text{Stab}_G(x)\]

For any $h \in G$,

\[\begin{aligned} &h \in \pi_x^{-1}(g \cdot x) \\\\ \iff &\pi_x(h) = g \cdot x \\\\ \iff &h \cdot x = g \cdot x \\\\ \iff &g^{-1} h \in \text{Stab}_G(x) &&(\star1)\\\\ \iff &h \in g\text{Stab}_G(x) &&(\star2) \end{aligned}\]

where:

  • $(\star 1)$ is justified by considering $g^{-1} h \cdot x = g^{-1} \cdot (h \cdot x) = g^{-1} \cdot (g \cdot x) = g^{-1} g \cdot x = x$.
  • $(\star 2)$ is justified by expanding definitions.

Hence the two sets are equal.

@Define what it means for a group action

\[\rho : G\times X \to X\]

to be faithful.

There does not exist a non-trivial $g \in G \setminus \{1\}$ such that $g \cdot x = x$ for all $x$, i.e. if some element fixes everything, that element is the identity.

Subquotients

Suppose $G$ is a group. @Define what it means for $Q$ to be a subquotient of $G$.

There exists $N \trianglelefteq H \le G$ such that

\[Q \cong H/N\]

Cauchy’s theorem

@State Cauchy’s theorem (for groups).

Suppose:

  • $G$ is a finite group
  • $p$ is a prime which divides $ \vert G \vert $

Then:

  • $G$ has an element of order $p$

Sylow theorems

Suppose:

  • $G$ is a group
  • $ \vert G \vert = p^n a$ where $p$ is a prime, $a, p$ are coprime, and $n \ge 0$

@State a result about the structure of $G$ in this case.
  • $G$ contains a subgroup $H$ such that $ \vert H \vert = p^n$
  • If $H, H’ \le G$ are two subgroups of size $p^n$, then they are conjugate to each other, i.e. there exists $g \in G$ such that $g^{-1} H g = H’$.

Fermat’s little theorem and Euler’s theorem

@State Fermat’s little theorem.

Suppose:

  • $p$ is a prime
  • $a$ is an integer coprime to $p$

Then:

\[a^{p-1} \equiv 1 \pmod p\]

@State Euler’s theorem.

Suppose:

  • $n \ge 2$
  • $a$ is an integer coprime to $p$
  • $\varphi$ is Euler’s totient function

Then:

\[a^{\varphi(n)} \equiv 1 \pmod n\]


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