[Abstract Algebra] Chapter 16 Flashcards
What is a group action?
An operation from G x X -> X
What property must a group action satisfy, relating to the group action itself?
(g1g2)x = g1(g2*x)
What property must a group action satisfy, relating to identities?
1g*x = x for all x
What type of group homomorphism is a group action equivalent to?
A homomorphism from G to S_X, the symmetric group on X
What is a stabilizer?
A stabilizer of a point x (in X) is the set of elements of G such that g * x = x
What is an orbit?
An equivalence class on X; x ~ y iff x = g * y for some g in G
What is the orbit-stabilizer theorem, in relation to left cosets?
Given any x in O, an orbit, and S = StabG(x), there exists a bijection between left cosets of S and O.
How is a stabilizer denoted?
StabG(x), where x is the point, and G is the group within the group operation
What is the orbit-stabilizer theorem, in relation to orders?
Given any x in O, an orbit, and S = StabG(x), |O||S|=|G|.
What is the orbit-stabilizer theorem, in relation to sizes?
Given any x in O, an orbit, and S = StabG(x), stabilizers of each x in O have the same size.
What is Burnside’s lemma?
The number of orbits in X is given by the arithmetic mean of the amount of fixed points for each g in G
What is a fixed point, in relation to Burnside’s lemma?
The set of points x in X such that g * x = x
What is the primary use of Burnside’s lemma?
To count orbits
What is the conjugation of elements?
An operation from G onto itself such that g : h = ghg^(-1)
What is the effect of conjugation of elements in a symmetric group?
It simply renames each element without effecting any properties of the elements
