Show the group action on (x, y) and, desribe the orbits.

[Edellaine]

Homework Statement

1 a) Prove that the group (nZ, +) acts on Z by a*g = a + g for all g in nZ and for all a in Z.
b)What are the orbits?
c)How many orbits are there? Do the set of orbits remind you of anything in number theory?

Homework Equations

not sure

The Attempt at a Solution

a) For e in (nZ, +), e = 0. so a*e = a+e = a+0 = a. So the identity of (nZ, +) is also the identity on Z.
For g, h in nZ, and for a in Z, a*(g+h) = a+(g+h) = (a+g)+h = (a*g)*h.

b, c) I know what the definition of what an orbit is (here for some a in Z, O_a = {a*g = a+g | g in nZ}) I'm not sure what the question is actually asking.

For c, would the amount of orbits be equal to the amount of elements in nZ, or 2n+1?

 

[Dick]

I don't think you are quite getting what the action is. Read the definition again. Two elements a and b are in the same orbit if a=b+k*n for some k in Z. Isn't that what it's saying? That really should ring a bell.