Generate the group table for Z_n

APL (10)

(Assuming ⎕IO=0. It works on ngn/apl by default, other APLs tend to need an ⎕IO←0 first.)

{⍵|∘.+⍨⍳⍵}

Explanation:

• ⍳⍵: the numbers [0..⍵)
• ∘.+⍨: create a sum table
• ⍵|: numbers in the table mod ⍵

GolfScript (13 chars)

I understand from your comment on Claudiu's answer that whitespace between the elements of a row is not necessary. On that understanding:

~.,{.n\(+}@(*

Online demo

Dissection:

~        Parse the input into an integer
.,       Duplicate it, turn the second into an array [0,...,n-1]
{        Loop: top of stack is the previous row
  .n\    Push a newline and a copy of the previous row
  (+     Rotate the first element to the end to get the new row
}@(*     Perform loop n-1 times

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If whitespace is necessary, for 20 chars:

~.,{.(+}@(*]{' '*n}/

Python 2, 66 bytes

def f(n):R=range(n);exec"print''.join(map(str,R));R+=R.pop(0),;"*n

Rotates the list by popping and re-appending.

Python 3, 53 bytes

def f(n):*R,=range(n);[print(*R[i:]+R[:i])for i in R]

Uses the same method as @mbomb007, but abusing print as a function.