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Displaying similar documents to “On the nilpotent residuals of all subalgebras of Lie algebras”

𝒟 n , r is not potentially nilpotent for n 4 r - 2

Yan Ling Shao, Yubin Gao, Wei Gao (2016)

Czechoslovak Mathematical Journal

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An n × n sign pattern 𝒜 is said to be potentially nilpotent if there exists a nilpotent real matrix B with the same sign pattern as 𝒜 . Let 𝒟 n , r be an n × n sign pattern with 2 r n such that the superdiagonal and the ( n , n ) entries are positive, the ( i , 1 ) ( i = 1 , , r ) and ( i , i - r + 1 ) ( i = r + 1 , , n ) entries are negative, and zeros elsewhere. We prove that for r 3 and n 4 r - 2 , the sign pattern 𝒟 n , r is not potentially nilpotent, and so not spectrally arbitrary.

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...

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...

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. ...

Enveloping algebras of Slodowy slices and the Joseph ideal

Alexander Premet (2007)

Journal of the European Mathematical Society

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Let G be a simple algebraic group over an algebraically closed field 𝕜 of characteristic 0, and 𝔤 = Lie G . Let ( e , h , f ) be an 𝔰 𝔩 2 -triple in 𝔤 with e being a long root vector in 𝔤 . Let ( · , · ) be the G -invariant bilinear form on 𝔤 with ( e , f ) = 1 and let χ 𝔤 * be such that χ ( x ) = ( e , x ) for all x 𝔤 . Let 𝒮 be the Slodowy slice at e through the adjoint orbit of e and let H be the enveloping algebra of 𝒮 ; see [31]. In this article we give an explicit presentation of H by generators and relations. As a consequence we deduce that H contains...

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...

Multiplicative Lie triple derivations on standard operator algebras

Bilal Ahmad Wani (2021)

Communications in Mathematics

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Let 𝒳 be a Banach space of dimension n > 1 and 𝔄 ( 𝒳 ) be a standard operator algebra. In the present paper it is shown that if a mapping d : 𝔄 𝔄 (not necessarily linear) satisfies d ( [ [ U , V ] , W ] ) = [ [ d ( U ) , V ] , W ] + [ [ U , d ( V ) ] , W ] + [ [ U , V ] , d ( W ) ] for all U , V , W 𝔄 , then d = ψ + τ , where ψ is an additive derivation of 𝔄 and τ : 𝔄 𝔽 I vanishes at second commutator [ [ U , V ] , W ] for all U , V , W 𝔄 . Moreover, if d is linear and satisfies the above relation, then there exists an operator S 𝔄 and a linear mapping τ from 𝔄 into 𝔽 I satisfying τ ( [ [ U , V ] , W ] ) = 0 for all U , V , W 𝔄 , such that d ( U ) = S U - U S + τ ( U ) for all U 𝔄 .