Chapter 4: Actions Flashcards
Define the concept of a group action.
What is the shorter notation?
How do we verbalise the relationship we’re describing?
An action ɑ of G on X is a function ɑ: G × X → X such that:
- For all 𝒙 ∈ X, we have that ɑ(𝙚, 𝒙) = 𝒙, i.e. the identity fixes all the elements of X
- For all 𝒈, 𝒉 ∈ G and 𝒙 ∈ X, we have that ɑ(𝒈, ɑ(𝒉, 𝒙)) = ɑ(𝒈𝒉, 𝒙)
Often ɑ(𝒈, 𝒙) is shortened to 𝒈(𝒙), 𝒈𝒙 or even 𝒈 ⋅ 𝒙.
We say that G acts on X, or that X is a G-set.
Give examples of group actions:
- Matrix groups
- Permutation groups
- Geometry
- A matrix group, like the subgroup of GL2(ℝ) that includes the cos/-sin/sin/cos “rotation matrices”, can act on the set of all 2d vectors, X, by left multiplication. Any matrix in this subgroup of GL2 will apply a rotation to any vector in X.
- [CONFIRM WITH TUTOR] Permutations in a group (e.g. S𝒏) act on the underlying set, X, by sending the “target” 𝒙 ∈ X to a “destination”/new value, with the identity perm sending 𝒙 to itself and perm1 of perm2 of 𝒙 being equal to (perm1perm2) of 𝒙 because the group operation for permutations is composition anyway? As a concrete example, if X is a set of ordered pairs e.g. Cartesian plane coordinates, with values from {1, …, 𝒏}, and σ1, σ2 ∈ S𝒏, then σ1 ∘ σ2 ∘ (1,2) = σ1σ2 ∘ (1,2)
- Given a geometric figure, φ, we can consider the group of all of its symmetries, Sym(φ). This group pretty much by definition acts on φ. For example, if P𝒏 is a regular polygon with 𝒏 sides then Sym(P𝒏) is the dihedral group Dih(2𝒏)/D𝒏, which acts on P𝒏 with half of its elements being rotations of P𝒏 and the other half being its reflections.
Define the conjugation action of a group on itself
[CONFIRM WITH TUTOR]
Instead of acting on a separate set, X, the group can act on its own set, i.e. X = G.
G acts on itself by conjugation.
This can happen in several ways. One is for G to act on itself by left multiplication::
- For 𝐞 in G and all 𝒙 in G, we have that 𝐞𝒙 = 𝒙
- Since group multiplication is associative, we have
(𝒈1𝒈2) ∘ 𝒙 = (𝒈1𝒈2)𝒙 = 𝒈1(𝒈2𝒙) = 𝒈1 ∘ (𝒈2 ∘ 𝒙)
[??? something about right multiplication? and then 𝒙𝒈/𝒈𝒙𝒈-1 is very important?]
State and prove Cayley’s theorem
Define the concept of an orbit and a stabiliser.
What are they each a subset of?
Are either a subgroup?
Let X be a G-set and let 𝒙 ∈ X.
The orbit of 𝒙 is:
orb(𝒙) = { 𝒈 ∘ 𝒙 : 𝒈 ∈ G }
__Informally, an orbit is the set of “all the places an element can go” e.g. in D3, ‹ r › = {e, r, r2} because any further operations with r will just take you back to e and then repeat the same set of rotations again.__
The stabiliser of 𝒙 is:
Stab(𝒙) = G𝒙 = { 𝒈 ∈ G: 𝒈 ∘ 𝒙 = 𝒙 }
__The stabiliser of 𝒙 is the set of all elements in G that send 𝒙 to itself.__
Notice that the orbit of 𝒙 is a subset of X, and the stabiliser of 𝒙 is a subset of G.
In fact, the stabiliser of 𝒙 is a subgroup of G.
State and prove the orbit-stabiliser theorem
TBC
Define the concept of the center of a group
TBC
State and prove Cauchy’s theorem
TBC
Classify groups of prime order
TBC
Classify groups of order 2p and p^2 for any prime p
TBC
Prove that conjugate elements have equal order
TBC
Classify the conjugacy classes of dihedral groups
TBC
Prove that conjugate matrices have equal traces
TBC
Prove that elements of the symmetric group are conjugate iff they have the same cycle type
TBC
Determine when elements of the alternating group are conjugate
TBC
