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A characterization of sequences with the minimum number of k-sums modulo k

Xingwu Xia, Yongke Qu, Guoyou Qian (2014)

Colloquium Mathematicae

Let G be an additive abelian group of order k, and S be a sequence over G of length k+r, where 1 ≤ r ≤ k-1. We call the sum of k terms of S a k-sum. We show that if 0 is not a k-sum, then the number of k-sums is at least r+2 except for S containing only two distinct elements, in which case the number of k-sums equals r+1. This result improves the Bollobás-Leader theorem, which states that there are at least r+1 k-sums if 0 is not a k-sum.

A characterization of symplectic groups related to Fermat primes

Behnam Ebrahimzadeh, Alireza K. Asboei (2021)

Commentationes Mathematicae Universitatis Carolinae

We proved that the symplectic groups PSp ( 4 , 2 n ) , where 2 2 n + 1 is a Fermat prime number is uniquely determined by its order, the first largest element orders and the second largest element orders.

A characterization of the linear groups L 2 ( p )

Alireza Khalili Asboei, Ali Iranmanesh (2014)

Czechoslovak Mathematical Journal

Let G be a finite group and π e ( G ) be the set of element orders of G . Let k ∈ π e ( G ) and m k be the number of elements of order k in G . Set nse ( G ) : = { m k : k ∈ π e ( G ) } . In fact nse ( G ) is the set of sizes of elements with the same order in G . In this paper, by nse ( G ) and order, we give a new characterization of finite projective special linear groups L 2 ( p ) over a field with p elements, where p is prime. We prove the following theorem: If G is a group such that | G | = | L 2 ( p ) | and nse ( G ) consists of 1 , p 2 - 1 , p ( p + ϵ ) / 2 and some numbers divisible by 2 p , where p is a prime greater than...

A characterization property of the simple group PSL 4 ( 5 ) by the set of its element orders

Mohammad Reza Darafsheh, Yaghoub Farjami, Abdollah Sadrudini (2007)

Archivum Mathematicum

Let ω ( G ) denote the set of element orders of a finite group G . If H is a finite non-abelian simple group and ω ( H ) = ω ( G ) implies G contains a unique non-abelian composition factor isomorphic to H , then G is called quasirecognizable by the set of its element orders. In this paper we will prove that the group P S L 4 ( 5 ) is quasirecognizable.

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