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A characterization of C 2 ( q ) where q > 5

Ali Iranmanesh, Behrooz Khosravi (2002)

Commentationes Mathematicae Universitatis Carolinae

The order of every finite group G can be expressed as a product of coprime positive integers m 1 , , m t such that π ( m i ) is a connected component of the prime graph of G . The integers m 1 , , m t are called the order components of G . Some non-abelian simple groups are known to be uniquely determined by their order components. As the main result of this paper, we show that the projective symplectic groups C 2 ( q ) where q > 5 are also uniquely determined by their order components. As corollaries of this result, the validities of a...

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.

A Hajós type result on factoring finite abelian groups by subsets. II

Keresztély Corrádi, Sándor Szabó (2010)

Commentationes Mathematicae Universitatis Carolinae

It is proved that if a finite abelian group is factored into a direct product of lacunary cyclic subsets, then at least one of the factors must be periodic. This result generalizes Hajós's factorization theorem.

A new characterization for the simple group PSL ( 2 , p 2 ) by order and some character degrees

Behrooz Khosravi, Behnam Khosravi, Bahman Khosravi, Zahra Momen (2015)

Czechoslovak Mathematical Journal

Let G be a finite group and p a prime number. We prove that if G is a finite group of order | PSL ( 2 , p 2 ) | such that G has an irreducible character of degree p 2 and we know that G has no irreducible character θ such that 2 p θ ( 1 ) , then G is isomorphic to PSL ( 2 , p 2 ) . As a consequence of our result we prove that PSL ( 2 , p 2 ) is uniquely determined by the structure of its complex group algebra.

A new characterization of Mathieu groups

Changguo Shao, Qinhui Jiang (2010)

Archivum Mathematicum

Let G be a finite group and nse ( G ) the set of numbers of elements with the same order in G . In this paper, we prove that a finite group G is isomorphic to M , where M is one of the Mathieu groups, if and only if the following hold: (1)  | G | = | M | , (2)  nse ( G ) = nse ( M ) .

A new characterization of Suzuki groups

Behnam Ebrahimzadeh, Reza Mohammadyari (2019)

Archivum Mathematicum

One of the important questions that remains after the classification of the finite simple groups is how to recognize a simple group via specific properties. For example, authors have been able to use graphs associated to element orders and to number of elements with specific orders to determine simple groups up to isomorphism. In this paper, we prove that Suzuki groups S z ( q ) , where q ± 2 q + 1 is a prime number can be uniquely determined by the order of group and the number of elements with the same order.

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