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{\bf P.~J.~Cameron, G.~R.~Omidi and B.~Tayfeh-Rezaie}
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{\bf $3$-Designs from PGL$(2,q)$}
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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.
\bye