# Notes - Galois Theory HT25, Groups

> Source: https://ollybritton.com/notes/uni/part-b/ht25/galois-theory/notes/groups/ · Updated: 2025-05-21 · Tags: uni, notes

- [Course - Galois Theory HT25](https://ollybritton.com/notes/uni/part-b/ht25/galois-theory/)
	- [Notes - Galois Theory HT25, Solvable groups](https://ollybritton.com/notes/uni/part-b/ht25/galois-theory/notes/solvable-groups/)
	- [Notes - Galois Theory HT25, Basic definitions](https://ollybritton.com/notes/uni/part-b/ht25/galois-theory/notes/basic-definitions/)
	- See also:
	- [Course - Groups and Group Actions HT23](https://ollybritton.com/notes/uni/prelims/ht23/groups/)
	- [Course - Groups and Group Actions TT23](https://ollybritton.com/notes/uni/prelims/tt23/groups/)
	- [Notes - Groups TT23, Group actions](https://ollybritton.com/notes/uni/prelims/tt23/groups/notes/group-actions/)
	- [Notes - Groups TT23, Orbits and stabilisers](https://ollybritton.com/notes/uni/prelims/tt23/groups/notes/orbits-and-stabilisers/)

### 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 $|G|$.

---

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 $|G|$.

#### 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 $|G|$

Then:

- $G$ has an element of order $p$

#### Sylow theorems
Suppose:

- $G$ is a group
- $|G| = p^n a$ 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 $|H| = p^n$
- 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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