• Title, Summary, Keyword: nilpotency class

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NILPOTENCY CLASSES OF RIGHT NILPOTENT CONGRUENCES

  • Jeong, Joo-Hee
    • Bulletin of the Korean Mathematical Society
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    • v.36 no.1
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    • pp.139-146
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    • 1999
  • It is known that a right nilpotent congruence $\beta$ on a finite algebra A is also left nilpotent [3]. The question on whether the left nilpotency class of $\beta$ in less than or equal to the right nilpotency class of $\beta$is still open. In this paper we find an upper limit for the left nilpotency class of $\beta$. In addition, under the assumption that 1 $\in$ typ{A}, we show that $(\beta]^k=[\beta)^k$ for all k$\geq$1. Thus the left and right nilpotency classes of $\beta$ are the same in this case.

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SOME PROPERTIES OF EVOLUTION ALGEBRAS

  • Camacho, L.M.;Gomez, J.R.;Omirov, B.A.;Turdibaev, R.M.
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.5
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    • pp.1481-1494
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    • 2013
  • The paper is devoted to the study of finite dimensional complex evolution algebras. The class of evolution algebras isomorphic to evolution algebras with Jordan form matrices is described. For finite dimensional complex evolution algebras the criterium of nilpotency is established in terms of the properties of corresponding matrices. Moreover, it is proved that for nilpotent $n$-dimensional complex evolution algebras the possible maximal nilpotency index is $1+2^{n-1}$.

GROUPS HAVING MANY 2-GENERATED SUBGROUPS IN A GIVEN CLASS

  • Gherbi, Fares;Trabelsi, Nadir
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.2
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    • pp.365-371
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    • 2019
  • If �� is a class of groups, denote by F�� the class of groups G such that for every $x{\in}G$, there exists a normal subgroup of finite index H(x) such that ${\langle}x,h{\rangle}{\in}$ �� for every $h{\in}H(x)$. In this paper, we consider the class F��, when �� is the class of nilpotent-by-finite, finite-by-nilpotent and periodic-by-nilpotent groups. We will prove that for the above classes �� we have that a finitely generated hyper-(Abelian-by-finite) group in F�� belongs to ��. As a consequence of these results, we prove that when the nilpotency class of the subgroups (or quotients) of the subgroups ${\langle}x,h{\rangle}$ are bounded by a given positive integer k, then the nilpotency class of the corresponding subgroup (or quotient) of G is bounded by a positive integer c depending only on k.

FINITE NON-NILPOTENT GENERALIZATIONS OF HAMILTONIAN GROUPS

  • Shen, Zhencai;Shi, Wujie;Zhang, Jinshan
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.6
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    • pp.1147-1155
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    • 2011
  • In J. Korean Math. Soc, Zhang, Xu and other authors investigated the following problem: what is the structure of finite groups which have many normal subgroups? In this paper, we shall study this question in a more general way. For a finite group G, we define the subgroup $\mathcal{A}(G)$ to be intersection of the normalizers of all non-cyclic subgroups of G. Set $\mathcal{A}_0=1$. Define $\mathcal{A}_{i+1}(G)/\mathcal{A}_i(G)=\mathcal{A}(G/\mathcal{A}_i(G))$ for $i{\geq}1$. By $\mathcal{A}_{\infty}(G)$ denote the terminal term of the ascending series. It is proved that if $G=\mathcal{A}_{\infty}(G)$, then the derived subgroup G' is nilpotent. Furthermore, if all elements of prime order or order 4 of G are in $\mathcal{A}(G)$, then G' is also nilpotent.

ON A GENERALIZATION OF THE MCCOY CONDITION

  • Jeon, Young-Cheol;Kim, Hong-Kee;Kim, Nam-Kyun;Kwak, Tai-Keun;Lee, Yang;Yeo, Dong-Eun
    • Journal of the Korean Mathematical Society
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    • v.47 no.6
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    • pp.1269-1282
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    • 2010
  • We in this note consider a new concept, so called $\pi$-McCoy, which unifies McCoy rings and IFP rings. The classes of McCoy rings and IFP rings do not contain full matrix rings and upper (lower) triangular matrix rings, but the class of $\pi$-McCoy rings contain upper (lower) triangular matrix rings and many kinds of full matrix rings. We first study the basic structure of $\pi$-McCoy rings, observing the relations among $\pi$-McCoy rings, Abelian rings, 2-primal rings, directly finite rings, and ($\pi-$)regular rings. It is proved that the n by n full matrix rings ($n\geq2$) over reduced rings are not $\pi$-McCoy, finding $\pi$-McCoy matrix rings over non-reduced rings. It is shown that the $\pi$-McCoyness is preserved by polynomial rings (when they are of bounded index of nilpotency) and classical quotient rings. Several kinds of extensions of $\pi$-McCoy rings are also examined.