Groups TT23, Orbits and stabilisers
Flashcards
Let $G$ be a group acting on $\Omega$. @Define the orbit of $x$, $\mathrm{Orb}(x)$.
Let $G$ be a group acting on $\Omega$. What is the stabiliser of $x$, $\mathrm{Stab}(x)$?
What does it mean for a group action to be transitive in a technical sense, and then in an intuitive sense?
For any $x \in \Omega$, $\mathrm{Orb}(x) = \Omega$, i.e. any element can be reached from any other element.
What is true about the structure of any stabiliser $\mathrm{Stab}(x)$ of an element with respect to any group action given by a group $G$?
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 equivalence relation do you consider on $\Omega$?
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.
@State the orbit stabiliser theorem.
Let $G$ be a finite group acting on a set $\Omega$. Let $x \in \Omega$. Then
\[ \vert G \vert = \vert \mathrm{Stab}(x) \vert \times \vert \mathrm{Orb}(x) \vert \]@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 $\mathrm{Stab}(eH) = H$ and $\mathrm{Orb}(eH) = G/H$, so
\[ \vert G/H \vert \times \vert H \vert = \vert G \vert \]Why do both $\mathrm{Orb}(x)$ and $\mathrm{Stab}(x)$ divide the order of the group?
- $\mathrm{Stab}(x)$ divides by Lagrange’s theorem
- $\mathrm{Orb}(x)$ 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?
given by
\[g\mathrm{Stab}(x) \mapsto g \cdot x\]What common technique allows you to find the order of symmetry groups such as the icosahedron or cube?
The Orbit-Stabiliser theorem, i.e.
\[ \vert G \vert = \vert \mathrm{Orb}(x) \vert \times \vert \mathrm{Stab}(x) \vert \]What bijection lets you prove the orbit-stabiliser theorem for an action $(\cdot) : G \times S \to S$?
given by
\[\phi(g\mathrm{Stab}(x)) = g \cdot x\]@Justify that the map
\[\phi: G/\mathrm{Stab}(x) \to \mathrm{Orb}(x)\]
given by
\[\phi(g\mathrm{Stab}(x)) = g \cdot x\]
is both well-defined and injective? (this is almost the proof of the orbit-stabiliser theorem)
Proofs
@Prove that in any group $G$ and for any group action acting on $\Omega$,
\[\mathrm{Stab}(x) \leqslant G\]
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
\[ \vert G \vert = \vert \mathrm{Stab}(x) \vert \times \vert \mathrm{Orb}(x) \vert \]
Todo (page 73, groups and group actions).