example of group action

Let a,b,c be integers and let [a,b,c] denote the mapping


Let G be the group of 2×2 matrices such that detA=±1AG. The substitution


leads to



a = aa112+ba11a21+ca212 (1)
b = 2aa11a12+2ca21a22+b(a11a22+a12a21)
c = aa122+ba12a22+ca222

So we define


to be the binary quadratic form with coefficients a,b,c of x2,xy,y2, respectively as in (1). Putting in A=1001 we have [a,b,c]A=[a,b,c] for any binary quadratic form [a,b,c]. Now let B be another matrix in G. We must show that


Set [a,b,c](AB):=[a′′,b′′,c′′]. So we have

a′′ = a(a11b11+a12b21)2+c(a21b11+a22b21)2+b(a11b11+a12b21)(a21b11+a22b21) (2)
= ab112+cb212+(2aa11a12+2ca21a22+b(a11a22+a12a21))b11b21
c′′ = a(a11b12+a12b22)2+c(a21b12+a22b22)2+b(a11b12+a12b22)(a21b12+a22b22) (3)
= ab122+cb222+(2aa11a12+2ca21a22+b(a11a22+a12a21))b12b22

as desired. For the coefficient b′′ we get

b′′ = 2a(a11b11+a12b21)(a11b12+a12b22)
+ 2c(a21b11+a22b21)(a21b12+a22b22)
+ b((a11b11+a12b21)(a21b12+a22b22)+(a11b12+a12b22)(a21b11+a22b21))

and by evaluating the factors of b11b12,b21b22, and b11b22+b21b12, it can be checked that


This shows that

[a′′,b′′,c′′]=[a,b,c]B (4)

and therefore [a,b,c](AB)=([a,b,c]A)B. Thus, (1) defines an action of G on the set of (integer) binary quadratic forms. Furthermore, the discriminantMathworldPlanetmathPlanetmathPlanetmath of each quadratic formMathworldPlanetmath in the orbit of [a,b,c] under G is b2-4ac.

Title example of group action
Canonical name ExampleOfGroupAction
Date of creation 2013-03-22 13:50:00
Last modified on 2013-03-22 13:50:00
Owner Thomas Heye (1234)
Last modified by Thomas Heye (1234)
Numerical id 11
Author Thomas Heye (1234)
Entry type Example
Classification msc 11E16
Classification msc 16W22
Classification msc 20M30