Putnam 1996 A4

Let $S$ be the set of ordered triples $(a, b, c)$ of distinct elements of a finite set $A$. Suppose that

1. $(a,b,c) \in S$ if and only if $(b,c,a) \in S$;

2. $(a,b,c) \in S$ if and only if $(c,b,a) \notin S$;

3. $(a,b,c)$ and $(c,d,a)$ are both in $S$ if and only if $(b,c,d)$ and $(d,a,b)$ are both in $S$.

Prove that there exists a one-to-one function $g$ from $A$ to $R$ such that $g(a) < g(b) < g(c)$ implies $(a,b,c) \in S$. Note: $R$ is the set of real numbers.