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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 ) .

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