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Displaying similar documents to “Thompson’s conjecture for the alternating group of degree 2 p and 2 p + 1

A variation of Thompson's conjecture for the symmetric groups

Mahdi Abedei, Ali Iranmanesh, Farrokh Shirjian (2020)

Czechoslovak Mathematical Journal

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Let G be a finite group and let N ( G ) denote the set of conjugacy class sizes of G . Thompson’s conjecture states that if G is a centerless group and S is a non-abelian simple group satisfying N ( G ) = N ( S ) , then G S . In this paper, we investigate a variation of this conjecture for some symmetric groups under a weaker assumption. In particular, it is shown that G Sym ( p + 1 ) if and only if | G | = ( p + 1 ) ! and G has a special conjugacy class of size ( p + 1 ) ! / p , where p > 5 is a prime number. Consequently, if G is a centerless group with N ( G ) = N ( Sym ( p + 1 ) ) , then...

Recognition of some families of finite simple groups by order and set of orders of vanishing elements

Maryam Khatami, Azam Babai (2018)

Czechoslovak Mathematical Journal

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Let G be a finite group. An element g G is called a vanishing element if there exists an irreducible complex character χ of G such that χ ( g ) = 0 . Denote by Vo ( G ) the set of orders of vanishing elements of G . Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let G be a finite group and M a finite nonabelian simple group such that Vo ( G ) = Vo ( M ) and | G | = | M | . Then G M . We answer in affirmative this conjecture for M = S z ( q ) , where q = 2 2 n + 1 and either q - 1 , q - 2 q + 1 or q + 2 q + 1 is a prime number, and M = F 4 ( q ) , where...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

Sum-product theorems and incidence geometry

Mei-Chu Chang, Jozsef Solymosi (2007)

Journal of the European Mathematical Society

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In this paper we prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P 1 , , P 4 , and Q 1 , , Q n 2 , if there are n ( 1 + δ ) / 2 many distinct lines between P i and Q j for all i , j , then P 1 , , P 4 are collinear. If the number of the distinct lines is < c n 1 / 2 then the cross ratio of the four points is algebraic. 2. Given c > 0 , there is δ > 0 such that for any P 1 , P 2 , P 3 2 noncollinear, and Q 1 , , Q n 2 , if there are c n 1 / 2 many distinct lines between P i and Q j for all i , j , then for any P 2 { P 1 , P 2 , P 3 } , we have δ n distinct lines between P and Q j . 3. Given...

C * -points vs P -points and P -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

A new characterization of symmetric group by NSE

Azam Babai, Zeinab Akhlaghi (2017)

Czechoslovak Mathematical Journal

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Let G be a group and ω ( G ) be the set of element orders of G . Let k ω ( G ) and m k ( G ) be the number of elements of order k in G . Let nse ( G ) = { m k ( G ) : k ω ( G ) } . Assume r is a prime number and let G be a group such that nse ( G ) = nse ( S r ) , where S r is the symmetric group of degree r . In this paper we prove that G S r , if r divides the order of G and r 2 does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.