Intersecting Families in the Alternating Group and Direct Product of Symmetric Groups

Abstract

Let $S_{n}$ denote the symmetric group on $[n]=\{1, \ldots, n\}$. A family $I \subseteq S_{n}$ is intersecting if any two elements of $I$ have at least one common entry. It is known that the only intersecting families of maximal size in $S_{n}$ are the cosets of point stabilizers. We show that, under mild restrictions, analogous results hold for the alternating group and the direct product of symmetric groups.

An Erdös--Ko--Rado Theorem for the Derangement Graph of ${PGL}_3(q)$ Acting on the Projective Plane

“…Recently there have been many papers questioning if the natural extension of the Erdős-Ko-Rado theorem holds for specific permutation groups G (see [2,9,11,13]). This means asking if the largest independent sets in the derangement graph Γ G are the cosets in G of the stabilizer of a point.…”

“…Consider the case when |{α, β, γ, δ}| = 3; we have seen that this implies that either α = δ or β = γ. From the symmetries in (9) and (10), we may assume that α = δ and that α, β, γ are non-collinear (otherwise all four points are collinear). Now, replacing α, β and γ if necessary by α h , β h and γ h for h ∈ G, we get α = ε 1 , β = ε 3 and γ = ε 2 .…”

An Erdős–Ko–Rado theorem for finite 2-transitive groups

“…Recently there have been many papers proving that the natural extension of the Erdős-Ko-Rado theorem holds for specific permutation groups G (see [1,13,26,28,29,33]) and there are also two papers, [2] and [3], that consider when the natural extension of the Erdős-Ko-Rado theorem holds for transitive and 2transitive groups. Again, this means asking if the largest intersecting sets in G are the cosets in G of the stabiliser of a point.…”

Characterization of intersecting families of maximum size in PSL(2,q)

“…Note that when q = 3, we have P SL(2, q) ∼ = A 4 , and the action of P SL(2, q) on the projective line P G (1, q) is equivalent to the (natural) action of A 4 on {1, 2, 3, 4}; in this case, it was pointed out in [16] that the set S = {(1), (123), (234)} (we are using cycle notation for permutations), is an intersecting family of maximum size in A 4 , but S is not a coset of any point stablizer. To prove Theorem 1 we apply a general method for solving Problem II for some 2-transitive groups.…”

“…where the first two terms in the right hand side of Equation (16) corresponds to x = 1 and x = −1. Furthermore, note that we have included a factor 1 2 in front of the last expression in Equation (16). This occurs because every element g λ having eigenvalues {1, x} also has eigenvalues {1, x −1 }.…”