TITLE: How can I prove that an order ("$<$" say) on $\mathbb Z_n$ cannot be defined?
QUESTION [6 upvotes]: I'm trying to show why it isn't possible to define an order of magnitude on $\mathbb Z_n$ (modular arithmetic) that satisfies the ordering properties of $\mathbb Z$. 
By letting addition to be $\oplus$ and multiplication $\otimes$, I know the following about $\mathbb Z_n$:

Closed under $\oplus$ and $\otimes$.
$\oplus$ and $\otimes$ are commutative.
$\oplus$ and $\otimes$ are associative.
$0$ is $\oplus$ identity and $1$ is $\otimes$ identity.
Cancellation: $(-a) \oplus a = 0$.
$$a \otimes (b \oplus c) = a \otimes b \oplus a \otimes c,$$
$$(b \oplus c) \otimes  a = b \otimes a \oplus c \otimes a .$$

How would I prove that at least one of the ordering properties of $\mathbb Z$ does not hold for $\mathbb Z_n$?
I will be definitely using the ordering properties of $\mathbb Z$:

exactly one of $a < b$ or $a = b$ or $a > b$ holds;
if $a < b$ and $b < c$, then $a < c$;
if $a < b$ then $a + c < b + c$;
if $a < b$ and $c > 0$ then $a \cdot c < b \cdot c$.

In a way I can see that one of these properties would collapse in $\mathbb Z_n$, but I cannot prove it. By contradiction perhaps? (start with $a < b$ and find end up with $a > b$?)
Any help much appreciated.

REPLY [7 votes]: It is a 1943 theorem of F.W. Levi that for a commutative group $(G,+)$ the following are 
equivalent:
(i) There exists a total ordering relation $\leq$ on $G$ which is compatible with the group law in the sense that for all $x_1,x_2,y_1,y_2 \in G$, $x_1 \leq y_1, \ x_2 \leq y_2 \implies x_1 + x_2 \leq y_1 + y_2$.  
(ii) $G$ is torsionfree: for all $x \in G$ and $n \in \mathbb{Z}^+$, $nx = 0 \implies x = 0$.  
A proof (and a citation to Levi's paper) is given in $\S 17.2$ of these notes.
Note that the implication (i) $\implies$ (ii) -- which is what was being asked about here -- is by far the easier one, and the argument I give for this is no different than those in the other answers.  (So this answer is more for those with a more general interest in ordered commutative groups.)