# Translate group definition into geometry system

I need to reword the definition of group (the four axioms: closure, associativity, identity and invertibility) to be lines and points of non-Euclidean geometry (the axiom system defined as geometry). And I just can't find an analogue for points and lines :(

I.e. for example, the set $\{0, 1, 2, 3\}$ relative to the operation $+_{\pmod 4}$ could be the model for the above system, but I simply cannot understand how to "draw" it using lines and points.

Many thanks in advance!

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I am confused. Why do you need to do this? –  Qiaochu Yuan Jan 25 '12 at 22:39
That's part of the homework I got. The course's subject is the axiom systems expressed in non-Euclidean geometry. What I'm asking above isn't exactly the homework question though, it's a prerequisite to solve another number of questions (which are much easier, if you don't have to build the model of lines and points...) –  wvxvw Jan 25 '12 at 22:57