$3$-Designs from PGL$(2,q)$

Abstract

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.

Vertex-Transitive Cubic Graphs of Square-Free Order

“…Since T is not regular, T αβ ∼ = Z 2 , and so G αβ ≤ T . Thus N G (G αβ ) ∼ = D 8 by [3], so N G (G αβ ) = G α . Then there are no x ∈ N G (G α ) and y ∈ N G (G αβ ) such that G α , x, y = G, a contradiction.…”

“…Then G = PSL(2, p) or PGL (2, p). Inspecting subgroups of G listed in [13,Chapter II,8.27] and [3], G does not have subgroups isomorphic to S 4 ×S 2 . Thus, G α is isomorphic to one of S 3 , D 12 , and S 4 .…”

“…It follows that all involutions in Aut are conjugate. Thus we may choose R such thatb 3 Assume that θ ∈ Aut has order 3. Note that Aut has an abelian Sylow 3-subgroup z,b 2 .…”

Some Results on 1‐Rotational Hamiltonian Cycle Systems

“…The forthcoming results exploit the following fairly known facts about PGL 2 (q) and PSL 2 (q) (for details we refer to [22] and [38]):…”

3-Spontaneous Emission Error Designs from PSL(2, q)or PGL(2, q)

“…These cases correspond to k = 4, 5, 6 in Theorem 3.2 by Cusack et al [10] and Omidi et al [22]; k = 4, 5, 6, 7 in Theorem 3.3 by Keranen et al [15] and Li et al [19,20]; k = 4, 5 in Theorem 3.4 by van Leijenhorst [26]. In particular, Cameron et al [7,8] considered all the cases of k ≡ 0, 1 (mod p) and p n ≡ 1, 3 (mod 4). Although the expressions of the lower bounds in Theorems 3.2-3.4 are lengthy, they generalize the above-mentioned known results and unify all the cases of 3 ≤ k ≤ q + 1.…”