Geodesic
\(3\)-designs from PGL\((2,q)\)
The electronic journal of combinatorics, Tome 13 (2006)

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Zbl EuDML
The group PGL$(2,q)$, $q=p^n$, $p$ an odd prime, is $3$-transitive on the projective line and therefore it can be used to construct $3$-designs. In this paper, we determine the sizes of orbits from the action of PGL$(2,q)$ on the $k$-subsets of the projective line when $k$ is not congruent to $0$ and 1 modulo $p$. Consequently, we find all values of $\lambda$ for which there exist $3$-$(q+1,k,\lambda)$ designs admitting PGL$(2,q)$ as automorphism group. In the case $p\equiv 3$ mod 4, the results and some previously known facts are used to classify 3-designs from PSL$(2,p)$ up to isomorphism.
DOI : 10.37236/1076
Classification : 05B05, 05B20
Mots-clés : \(t\)-design, automorphism group, Möbius function, linear group 
P. J. Cameron; G. R. Omidi; B. Tayfeh-Rezaie. \(3\)-designs from PGL\((2,q)\). The electronic journal of combinatorics, Tome 13 (2006). doi: 10.37236/1076
@article{10_37236_1076,
     author = {P. J. Cameron and G. R. Omidi and B. Tayfeh-Rezaie},
     title = {\(3\)-designs from {PGL\((2,q)\)}},
     journal = {The electronic journal of combinatorics},
     year = {2006},
     volume = {13},
     doi = {10.37236/1076},
     zbl = {1097.05007},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1076/}
}
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