On the maximum intersecting sets of the general semilinear group of degree 2

2024/02/27 by Maleki, Roghayeh, Razafimahatratra, Andriaherimanana Sarobidy
#05C35 #05C69 #20B05 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.17687

Abstract

Let p be a prime and q = p^k. A subset F ⊂ ΓL₂(q) is intersecting if any two semilinear transformations in F agree on some non-zero vector in \mathbbF_q². We show that any intersecting set of ΓL₂(q) is of size at most that of a stabilizer of a non-zero vector, and we characterize the intersecting sets of this size. Our proof relies on finding a subgraph which is a lexicographic product in the derangement graph of ΓL₂(q) in its action on non-zero vectors of \mathbbF_q². This method is also applied to give a new proof that the only maximal intersecting sets of GL₂(q) are the maximum intersecting sets.

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