Notes - Groups TT23, Orbits and stabilisers

Created: May 08, 2023
Updated: October 17, 2024

Flashcards

Let $G$ be a group acting on $\Omega$. Define the orbit of $x$, $\text{Orb}(x)$.
\[\text{Orb}(x) = \\{g \cdot x : \forall g \in G\\}\]

Let $G$ be a group acting on $\Omega$. What is the stabiliser of $x$, $\text{Stab}(x)$?
\[\text{Stab}(x) = \\{g \in G : g \cdot x = x\\}\]

What does it mean for a group action to be transitive in a technical sense, and then in an intuitive sense?

If for any $x \in \Omega$, $\text{Orb}(x) = \Omega$. Any element can be reached from any other element.

What is true about the structure of any stabiliser $\text{Stab}(x)$ of an element with respect to any group action given by a group $G$?
\[\text{Stab}(x) \leqslant G\]

What is true about the orbits of an action in relation to the set $\Omega$?

The orbits of an action partition the set $\Omega$.

When proving that the orbits of an action partition a set, what equilvance relation do you consider on $\Omega$?
\[s \sim t\]

if and only if $\exists g \in G$ such that $g \cdot s = t$.

What’s a quick proof that the centraliser of an element $x \in G$ defined by

\[C_G(x) = \\{g \in G : gx = xg\\}\]

is a subgroup?

This is the stabiliser of $x$ under the group action of conjugation, hence a subgroup.

Can you state the orbit stabiliser theorem?

Let $G$ be a finite group acting on a set $\Omega$. Let $x \in \Omega$. Then

\[|G| = |\text{Stab}(x)| \times |\text{Orb}(x)|\]

Can you quickly prove Lagrange’s theorem by using the Orbit-Stabiliser theorem?

Let $G$ be a group and let $H \leqslant G$. Then let $G$ act on $G/H$ by

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

Then $\text{Stab}(eH) = H$ and $\text{Orb}(eH) = G/H$, so

\[|G/H| \times |H| = |G|\]

Why do both $\text{Orb}(x)$ and $\text{Stab}(x)$ divide the order of the group?
  • $\text{Stab}(x)$ divides by Lagrange’s theorem
  • $\text{Orb}(x)$ divides by the Orbit-Stabiliser theorem

When proving the Orbit-Stabiliser theorem, what bijection do you consider where the proof then follows from Lagrange’s theorem?
\[\phi: G/\text{Stab}(x) \to \text{Orb}(x)\]

given by

\[g\text{Stab}(x) \mapsto g \cdot x\]

What common technique allows you to find the order of symmetry groups such as the icosahedron or cube?

The Orbit-Stabiliser theorem, i.e.

\[|G| = |\text{Orb}(x)|\times|\text{Stab}(x)|\]

What bijection lets you prove the orbit-stabiliser theorem for an action $(\cdot) : G \times S \to S$?
\[\phi: G/\text{Stab}(x) \to \text{Orb}(x)\]

given by

\[\phi(g\text{Stab}(x)) = g \cdot x\]

Can you quickly justify that the map

\[\phi: G/\text{Stab}(x) \to \text{Orb}(x)\]

given by

\[\phi(g\text{Stab}(x)) = g \cdot x\]

is both well-defined and injective? (this is almost the proof of the orbit-stabiliser theorem)
\[\begin{aligned} g\text{Stab}(x) = h\text{Stab}(x) &\iff h^{-1}g \in \text{Stab}(x) \\\\ &\iff h^{-1}g\cdot x = x \\\\ &\iff g\cdot x = h \cdot x \end{aligned}\]

Proofs

Prove that in any group $G$ and for any group action acting on $\Omega$,

\[\text{Stab}(x) \leqslant G\]

Todo (page 71, groups and group actions)

Prove that in any group $G$ and for any group action acting on $\Omega$, the orbits of elements in $\Omega$ partition the set.

Todo (page 70, groups and group actions)

Prove the Orbit-Stabiliser theorem:

Let $G$ be a finite group acting on a set $\Omega$. Let $x \in \Omega$. Then

\[\vert G \vert = \vert \text{Stab}(x) \vert \times \vert \text{Orb}(x) \vert\]

Todo (page 73, groups and group actions).



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