Some Erdös-Ko-Rado results for linear and affine groups of degree two

Meagher, Karen; Sarobidy, Razafimahatratra, Andriaherimanana

doi:10.26493/2590-9770.1494.1e4

Cited by 4 publications

(3 citation statements)

References 25 publications

(46 reference statements)

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“…It is well known (see [2] or [3], for example) that, since G n contains a Singer cycle as a regular subgroup, the size of every 1-intersecting set in G n is at most the expression given in (1•1) for t = 1. Meagher and Razafimahatratra [21] have shown that, if Y is a 1-intersecting set of size q 2 − q in G 2 , then 1 Y is in the span of the characteristic vectors of the 1-cosets. We prove a corresponding result for all t and n for which n is sufficiently large compared to t. THEOREM 1•1.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Intersection theorems for finite general linear groups

ERNST¹,

SCHMIDT²

2023

Math. Proc. Camb. Phil. Soc.

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A subset Y of the general linear group $\text{GL}(n,q)$ is called t-intersecting if $\text{rk}(x-y)\le n-t$ for all $x,y\in Y$ , or equivalently x and y agree pointwise on a t-dimensional subspace of $\mathbb{F}_q^n$ for all $x,y\in Y$ . We show that, if n is sufficiently large compared to t, the size of every such t-intersecting set is at most that of the stabiliser of a basis of a t-dimensional subspace of $\mathbb{F}_q^n$ . In case of equality, the characteristic vector of Y is a linear combination of the characteristic vectors of the cosets of these stabilisers. We also give similar results for subsets of $\text{GL}(n,q)$ that intersect not necessarily pointwise in t-dimensional subspaces of $\mathbb{F}_q^n$ and for cross-intersecting subsets of $\text{GL}(n,q)$ . These results may be viewed as variants of the classical Erdős–Ko–Rado Theorem in extremal set theory and are q-analogs of corresponding results known for the symmetric group. Our methods are based on eigenvalue techniques to estimate the size of the largest independent sets in graphs and crucially involve the representation theory of $\text{GL}(n,q)$ .

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Section: Introduction and Resultsmentioning

confidence: 99%

Intersection theorems for finite general linear groups

ERNST¹,

SCHMIDT²

2023

Math. Proc. Camb. Phil. Soc.

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“…Meagher and Sin [23] recently showed that all finite 2-transitive groups have the EKR-module property. However, the strict-EKR property does not hold for permutations groups in general; recently, Meagher and Razafimahatratra [22] have shown that the general linear group GL(2, q) is such a counterexample. We remark that our results are of similar flavor, although in our context of Peisert-type graphs, the corresponding vector space W does not carry a natural module structure.…”

Section: Ekr-type Resultsmentioning

confidence: 99%

The EKR-Module Property of Pseudo-Paley Graphs of Square Order

Shamil

Goryaĭnov

Lin

et al. 2022

Electron. J. Combin.

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We prove that a family of pseudo-Paley graphs of square order obtained from unions of cyclotomic classes satisfies the Erdős-Ko-Rado (EKR) module property, in a sense that the characteristic vector of each maximum clique is a linear combination of characteristic vectors of canonical cliques. This extends the EKR-module property of Paley graphs of square order and solves a problem proposed by Godsil and Meagher. Different from previous works, which heavily rely on tools from number theory, our approach is purely combinatorial in nature. The main strategy is to view these graphs as block graphs of orthogonal arrays, which is of independent interest.

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“…In [17], a measure for intersecting sets in transitive groups was introduced. Formally, the intersection density of a transitive group G ≤ Sym(Ω) is the rational number ρ(G) := max {ρ(F ) | F ⊂ G is intersecting} , where ρ(F ) := |F | |Gω| for any intersecting set F and G ω is the point stabilizer of ω ∈ Ω in G. The study of intersection density in transitive groups has recently drawn the attention of many researchers [1,11,12,13,18,19,21,22,20,24,26].…”

Section: Introductionmentioning

confidence: 99%

Intersection density of imprimitive groups of degree $pq$

Behajaina¹,

Roghayeh²,

Sarobidy³

2022

Preprint

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A subset F of a finite transitive group G ≤ Sym(Ω) is intersecting if any two elements of F agree on an element of Ω. The intersection density of G is the number ρ(G) = max |F | |G|/|Ω| | F ⊂ G is intersecting . Recently, Hujdurović et al. [11] disproved a conjecture of Meagher et al. [21, Conjecture 6.6 (3)]by constructing equidistant cyclic codes which yield transitive groups of degree pq, where p = q k −1 q−1 and q are odd primes, and whose intersection density equal to q.In this paper, we use the cyclic codes given by Hujdurović et al. and their permutation automorphisms to construct a family of transitive groups G of degree pq with ρ(G) = q k , whenever k < q < p = q k −1 q−1 are odd primes. Moreover, we extend their construction using cyclic codes of higher dimension to obtain a new family of transitive groups of degree a product of two odd primes q < p = q k −1 q−1 , and whose intersection density are equal to q. Finally, we prove that if G ≤ Sym(Ω) of degree a product of two arbitrary odd primes p > q and ω∈Ω Gω is a proper subgroup, then ρ(G) ∈ {1, q}.

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Some Erdös-Ko-Rado results for linear and affine groups of degree two

Cited by 4 publications

References 25 publications

Intersection theorems for finite general linear groups

Intersection theorems for finite general linear groups

The EKR-Module Property of Pseudo-Paley Graphs of Square Order

Intersection density of imprimitive groups of degree $pq$

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