Isomorphism testing in STS(13) What is the simplest isomorphism invariant which can distinguish between the two non-isomorphic Steiner triple systems on $13$ points?
Train structure and cycle structure, as described here, do the job, but is there a simple way? Maybe via $p$-ranks or something like that?
 A: There is a construction given in Theorem II.2.2.10 of the Handbook of 
Combinatorial designs for a STS of order a prime of form $6t+1$. It is clear 
that the resulting STS has a cyclic automorphism acting fixed-point-freely on 
points. The other STS of order 13 has a full automorphism group isomorphic to $S_{3}$. 
So the automorphism group could be a good invariant, depending on what you mean 
by simple. 
Note that it is possible to label the blocks of the designs so that they differ 
in just six blocks (they differ by a switching operation). So it seems unlikely that a simple test which looks at local properties of an STS will be able to distinguish the two triple systems, though some less computationally expensive invariant may suffice in this small case. 
A: Take the 26-vertex graphs whose vertices are the blocks and where two vertices are joined by an edge if the corresponding blocks have a vertex in common.  These two graphs differ in many easily measured ways, for example having different numbers of induced cycles and different numbers of maximal independent sets.  I suspect (but am too lazy to prove) that the two STSs have different numbers of Fano subplanes.
A: Actually, I now know that perhaps the easiest way is to count the number of Pasch configurations - one of the STS(13) has thirteen of them and the other eight. (source)
