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## Homework Statement

**1**a) Prove that the group (n

**Z**, +) acts on

**Z**by a*g = a + g for all g in n

**Z**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 (n

**Z**, +), e = 0. so a*e = a+e = a+0 = a. So the identity of (n

**Z**, +) is also the identity on

**Z**.

For g, h in n

**Z**, 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 n

**Z**}) 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 n

**Z**, or 2n+1?