Displaying similar documents to “ 𝒟 n , r is not potentially nilpotent for n ≥ 4 r - 2 ”

A new family of spectrally arbitrary ray patterns

Yinzhen Mei, Yubin Gao, Yan Ling Shao, Peng Wang (2016)

Czechoslovak Mathematical Journal

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An n × n ray pattern 𝒜 is called a spectrally arbitrary ray pattern if the complex matrices in Q ( 𝒜 ) give rise to all possible complex polynomials of degree n . In a paper of Mei, Gao, Shao, and Wang (2014) was proved that the minimum number of nonzeros in an n × n irreducible spectrally arbitrary ray pattern is 3 n - 1 . In this paper, we introduce a new family of spectrally arbitrary ray patterns of order n with exactly 3 n - 1 nonzeros.

Retracts that are kernels of locally nilpotent derivations

Dayan Liu, Xiaosong Sun (2022)

Czechoslovak Mathematical Journal

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Let k be a field of characteristic zero and B a k -domain. Let R be a retract of B being the kernel of a locally nilpotent derivation of B . We show that if B = R ⊕ I for some principal ideal I (in particular, if B is a UFD), then B = R [ 1 ] , i.e., B is a polynomial algebra over R in one variable. It is natural to ask that, if a retract R of a k -UFD B is the kernel of two commuting locally nilpotent derivations of B , then does it follow that B ≅ R [ 2 ] ? We give a negative answer to this question. The interest in...

On the nilpotent residuals of all subalgebras of Lie algebras

Wei Meng, Hailou Yao (2018)

Czechoslovak Mathematical Journal

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Let 𝒩 denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra L over an arbitrary field 𝔽 , there exists a smallest ideal I of L such that L / I ∈ 𝒩 . This uniquely determined ideal of L is called the nilpotent residual of L and is denoted by L 𝒩 . In this paper, we define the subalgebra S ( L ) = ⋂ H ≤ L I L ( H 𝒩 ) . Set S 0 ( L ) = 0 . Define S i + 1 ( L ) / S i ( L ) = S ( L / S i ( L ) ) for i ≥ 1 . By S ∞ ( L ) denote the terminal term of the ascending series. It is proved that L = S ∞ ( L ) if and only if L 𝒩 is nilpotent. In addition, we investigate the basic properties of a...

The p -nilpotency of finite groups with some weakly pronormal subgroups

Jianjun Liu, Jian Chang, Guiyun Chen (2020)

Czechoslovak Mathematical Journal

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For a finite group G and a fixed Sylow p -subgroup P of G , Ballester-Bolinches and Guo proved in 2000 that G is p -nilpotent if every element of P ∩ G ' with order p lies in the center of N G ( P ) and when p = 2 , either every element of P ∩ G ' with order 4 lies in the center of N G ( P ) or P is quaternion-free and N G ( P ) is 2 -nilpotent. Asaad introduced weakly pronormal subgroup of G in 2014 and proved that G is p -nilpotent if every element of P with order p is weakly pronormal in G and when p = 2 , every element of P with...

A note on infinite a S -groups

Reza Nikandish, Babak Miraftab (2015)

Czechoslovak Mathematical Journal

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Let G be a group. If every nontrivial subgroup of G has a proper supplement, then G is called an a S -group. We study some properties of a S -groups. For instance, it is shown that a nilpotent group G is an a S -group if and only if G is a subdirect product of cyclic groups of prime orders. We prove that if G is an a S -group which satisfies the descending chain condition on subgroups, then G is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group...

On a generalization of a theorem of Burnside

Jiangtao Shi (2015)

Czechoslovak Mathematical Journal

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A theorem of Burnside asserts that a finite group G is p -nilpotent if for some prime p a Sylow p -subgroup of G lies in the center of its normalizer. In this paper, let G be a finite group and p the smallest prime divisor of | G | , the order of G . Let P ∈ Syl p ( G ) . As a generalization of Burnside’s theorem, it is shown that if every non-cyclic p -subgroup of G is self-normalizing or normal in G then G is solvable. In particular, if P ≇ 〈 a , b | a p n - 1 = 1 , b 2 = 1 , b - 1 a b = a 1 + p n - 2 〉 , where n ≥ 3 for p > 2 and n ≥ 4 for p = 2 , then G is p -nilpotent or p -closed. ...

A property which ensures that a finitely generated hyper-(Abelian-by-finite) group is finite-by-nilpotent

Fares Gherbi, Nadir Trabelsi (2024)

Czechoslovak Mathematical Journal

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Let 𝔐 be the class of groups satisfying the minimal condition on normal subgroups and let Ω be the class of groups of finite lower central depth, that is groups G such that γ i ( G ) = γ i + 1 ( G ) for some positive integer i . The main result states that if G is a finitely generated hyper-(Abelian-by-finite) group such that for every x ∈ G , there exists a normal subgroup H x of finite index in G satisfying 〈 x , x h 〉 ∈ 𝔐 Ω for every h ∈ H x , then G is finite-by-nilpotent. As a consequence of this result, we prove that a finitely generated...

Strongly 2-nil-clean rings with involutions

Huanyin Chen, Marjan Sheibani Abdolyousefi (2019)

Czechoslovak Mathematical Journal

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A * -ring R is strongly 2-nil- * -clean if every element in R is the sum of two projections and a nilpotent that commute. Fundamental properties of such * -rings are obtained. We prove that a * -ring R is strongly 2-nil- * -clean if and only if for all a ∈ R , a 2 ∈ R is strongly nil- * -clean, if and only if for any a ∈ R there exists a * -tripotent e ∈ R such that a - e ∈ R is nilpotent and e a = a e , if and only if R is a strongly * -clean SN ring, if and only if R is abelian, J ( R ) is nil and R / J ( R ) is * -tripotent. Furthermore, we explore...