On intersection density of transitive groups of degree a product of two odd primes
Section snippets
Introductory remarks
Throughout this paper p and q will always denote prime numbers with .
Let be a permutation group acting on a set V, where denotes the full symmetric group on V. Two elements are said to be intersecting if for some . Furthermore, a subset of G is an intersecting set if every pair of elements of is intersecting. The intersection density of the intersecting set is defined to be the quotient where is the point stabilizer of
Hierarchy of transitive groups of degree pq
Let G be a transitive permutation group G acting on a set V. A partition of V is called G-invariant if the elements of G permute the parts, the so called blocks of , setwise. If the trivial partitions and are the only G-invariant partitions of V, then G is primitive, and is imprimitive otherwise. In the latter case a corresponding nontrivial G-invariant partition will be referred to as a complete imprimitivity block system of G. We say that G is doubly transitive if given any
Cyclic codes
Let m be a positive integer, r a power of a prime, and the finite field with r elements. The polynomial has no repeated factors (which are irreducible over ) if and only if r and m are relatively prime, i.e. (see [5, Exercise 201]), which we assume in this section.
Let be the m-dimensional vector space over formed by all row vectors with entries in . Let C be a linear code, that is, a k-dimensional vector subspace in . A linear code C
A family of groups with intersection density q
We start this section by proving Theorem 1.2. Recall from the introductory section that to every cyclic code C of length m over we can associate an imprimitive permutation group acting on .
Proof of Theorem 1.2 Let C be a nonzero cyclic code of length m over such that no codeword has maximal Hamming weight m, and let be the permutation group associated with C. Let K be the subgroup of generated by . Observe that defined by is an isomorphism between the additive group
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2023, Journal of Algebraic CombinatoricsBLOCK DESIGNS, PERMUTATION GROUPS AND PRIME VALUES OF POLYNOMIALS
2023, Trudy Instituta Matematiki i Mekhaniki UrO RAN
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The work of Ademir Hujdurović is supported in part by the Slovenian Research Agency (research program P1-0404 and research projects N1-0062, J1-9110, N1-0102, J1-1691, J1-1694, J1-1695, N1-0140, N1-0159, J1-2451 and N1-0208).
- 2
The work of Klavdija Kutnar is supported in part by the Slovenian Research Agency (research program P1-0285 and research projects N1-0062, J1-9110, J1-9186, J1-1695, J1-1715, N1-0140, J1-2451, J1-2481 and N1-0209).
- 3
The work of Bojan Kuzma is supported in part by the Slovenian Research Agency (research program P1-0285 and research project N1-0210).
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The work of Dragan Marušič is supported in part by the Slovenian Research Agency (I0-0035, research program P1-0285 and research projects N1-0062, J1-9108, J1-1694, J1-1695, N1-0140 and J1-2451).