# The Cyclically Resolvable Steiner Triple Systems of Order 57
## 2 The Action of the Automorphism Group of a Cyclic STS(57)
### Point-Cyclic Resolutions of STS(57) and Their Parallel Class Orbits Under \(G_{57}\)  
According to [PERSON] and [PERSON] [29] there are three types of point-cyclic KTS(\(v\))s:  
* \(T_{1}\)--the short block orbit under \(G_{v}\) is a parallel class of the resolution;
* \(T_{2}\)--each block from the short block orbit is in a different parallel class of the resolution;
* \(T_{3}\)--the resolution has parallel classes with more than one block from the short block orbit. Under \(G_{19}\) the cyclic STS(57) can have at most \(19\) block orbits which are parallel classes. \(G_{19}\) is generated by \(g^{3}\) which acts on the points as  
\[(0,3,6,\ldots,51,54)(1,4,7,\ldots,52,55)(2,5,8,\ldots,53,56),\]  
i.e., under \(G_{19}\) there are three point orbits \(o_{1}=\{0,3,6,\ldots,51,54\}\), \(o_{2}=\{1,4,7,\ldots,52,55\}\), and \(o_{3}=\{2,5,8,\ldots,53,56\}\). If a block \(B\) of the STS(57) has two points \(a\) and \(b\) from one and the same orbit \(o_{i}\), there is an automorphism from \(G_{19}\) mapping \(a\) to \(b\) and \(B\) to \(B^{\prime}\). Therefore the point \(b\) is in two blocks \(B\) and \(B^{\prime}\) of the corresponding block orbit and that is why this block orbit is not a parallel class. If a block orbit under \(G_{19}\) is a parallel class, all its blocks contain one point from each point orbit under \(G_{19}\). In each block orbit which is a parallel class there is a block containing point 0, a point \(a\) from \(o_{2}\), and a point \(b\) from \(o_{3}\). There are 19 different ways to choose the point \(a\) from \(o_{2}\). Since the pair of points \((0,a)\) should be in exactly one block of the Steiner triple system, there can be at most 19 blocks containing point 0 and a point \(a\) from \(o_{2}\) and therefore there can be not more than 19 block orbits under \(G_{19}\) which are parallel classes. All the point-cyclic resolutions of a cyclic STS(57) are of type \(T_{2}\) and under \(G_{57}\) their parallel classes are in \(1\) orbit of length 19 and \(3\) orbits of length 3. The number of parallel classes is 28. That is why a class orbit of length 57 is not possible, class orbits should be of lengths 1 (the classes are fixed by \(G_{57}\)), 3 (the classes are fixed by \(G_{19}\)) or 19 (the classes are fixed by \(G_{3}\)). There is only one parallel class which is fixed by \(G_{57}\), namely the short block orbit. That is why there can be at most one parallel class orbit of length 1. Therefore there are only two possibilities for the parallel class orbits of a point-cyclic KTS(57), namely either 9 class orbits of length 3 and one of length 1, or 3 class orbits of length 3 and one of length 19. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Length** & **1** & **3** & **19** & **57** \\ \hline \(G_{3}\) & 19 & 171 & & \\ \(G_{19}\) & & & 28 & \\ \(G_{57}\) & & & 1 & 9 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Block orbits of a cyclic STS(57) under \(G_{57}\) and its subgroups. A resolution of type \(T_{1}\) should consist of one class orbit of length 1 and nine class orbits of length 3. In this case \(G_{19}\) should fix each parallel class, namely each parallel class should consist of the blocks of one block orbit under \(G_{19}\), and therefore all the 28 block orbits under \(G_{19}\) should be parallel classes. That is why by Lemma 1 a resolution of type \(T_{1}\) is not possible. A resolution of type \(T_{2}\) or \(T_{3}\) should consist of one class orbit of length 19 and three class orbits of length 3. In this case \(G_{3}\) should fix the parallel classes from the orbit of length 19. Since a parallel class has 19 blocks, \(G_{3}\) can only fix it if it contains at least one block which is fixed by \(G_{3}\). Since \(G_{3}\) fixes exactly 19 blocks (the blocks of the short orbit) each class of the class orbit of length 19 should contain exactly one of the 19 blocks of the short block orbit. Therefore all point-cyclic resolutions of STS(57)s are of type \(T_{2}\).