• 제목/요약/키워드: Finite groups

검색결과 365건 처리시간 0.024초

OUTER AUTOMORPHISM GROUPS OF POLYGONAL PRODUCTS OF CERTAIN CONJUGACY SEPARABLE GROUPS

  • Kim, Goan-Su;Tang, Chi Yu
    • 대한수학회지
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    • 제45권6호
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    • pp.1741-1752
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    • 2008
  • Grossman [7] showed that certain cyclically pinched 1-relator groups have residually finite outer automorphism groups. In this paper we prove that tree products of finitely generated free groups amalgamating maximal cyclic subgroups have residually finite outer automorphism groups. We also prove that polygonal products of finitely generated central subgroup separable groups amalgamating trivial intersecting central subgroups have residually finite outer automorphism groups.

GROUPS HAVING MANY 2-GENERATED SUBGROUPS IN A GIVEN CLASS

  • Gherbi, Fares;Trabelsi, Nadir
    • 대한수학회보
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    • 제56권2호
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    • pp.365-371
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    • 2019
  • If 𝖃 is a class of groups, denote by F𝖃 the class of groups G such that for every $x{\in}G$, there exists a normal subgroup of finite index H(x) such that ${\langle}x,h{\rangle}{\in}$ 𝖃 for every $h{\in}H(x)$. In this paper, we consider the class F𝖃, when 𝖃 is the class of nilpotent-by-finite, finite-by-nilpotent and periodic-by-nilpotent groups. We will prove that for the above classes 𝖃 we have that a finitely generated hyper-(Abelian-by-finite) group in F𝖃 belongs to 𝖃. 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 p-GROUPS ALL OF WHOSE SUBGROUPS OF CLASS 2 ARE GENERATED BY TWO ELEMENTS

  • Li, Pujin;Zhang, Qinhai
    • 대한수학회지
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    • 제56권3호
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    • pp.739-750
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    • 2019
  • We proved that finite p-groups in the title coincide with finite p-groups all of whose non-abelian subgroups are generated by two elements. Based on the result, finite p-groups all of whose subgroups of class 2 are minimal non-abelian (of the same order) are classified, respectively. Thus two questions posed by Berkovich are solved.

POLYGONAL PRODUCTS OF RESIDUALLY FINITE GROUPS

  • Wong, Kok-Bin;Wong, Peng-Choon
    • 대한수학회보
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    • 제44권1호
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    • pp.61-71
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    • 2007
  • A group G is called cyclic subgroup separable for the cyclic subgroup H if for each $x\;{\in}\;G{\backslash}H$, there exists a normal subgroup N of finite index in G such that $x\;{\not\in}\;HN$. Clearly a cyclic subgroup separable group is residually finite. In this note we show that certain polygonal products of cyclic subgroup separable groups amalgamating normal subgroups are again cyclic subgroup separable. We then apply our results to polygonal products of polycyclic-by-finite groups and free-by-finite groups.

OUTER AUTOMORPHISM GROUPS OF CERTAIN POLYGONAL PRODUCTS OF GROUPS

  • Kim, Goan-Su
    • 대한수학회보
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    • 제45권1호
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    • pp.45-52
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    • 2008
  • We show that certain polygonal products of any four groups, amalgamating central subgroups with trivial intersections, have Property E. Using this result, we derive that outer automorphism groups of polygonal products of four polycyclic-by-finite groups, amalgamating central subgroups with trivial intersections, are residually finite.

RESIDUAL p-FINITENESS OF CERTAIN HNN EXTENSIONS OF FREE ABELIAN GROUPS OF FINITE RANK

  • Chiew Khiam Tang;Peng Choon Wong
    • 대한수학회보
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    • 제61권3호
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    • pp.785-796
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    • 2024
  • Let p be a prime. A group G is said to be residually p-finite if for each non-trivial element x of G, there exists a normal subgroup N of index a power of p in G such that x is not in N. In this note we shall prove that certain HNN extensions of free abelian groups of finite rank are residually p-finite. In addition some of these HNN extensions are subgroup separable. Characterisations for certain one-relator groups and similar groups including the Baumslag-Solitar groups to be residually p-finite are proved.

Residual P-Finiteness of Certain Generalized Free Products of Nilpotent Groups

  • Kim, Goan-Su;Lee, Young-Mi;McCarron, James
    • Kyungpook Mathematical Journal
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    • 제48권3호
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    • pp.495-502
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    • 2008
  • We show that free products of finitely generated and residually p-finite nilpotent groups, amalgamating p-closed central subgroups are residually p-finite. As a consequence, we are able to show that generalized free products of residually p-finite abelian groups are residually p-finite if the amalgamated subgroup is closed in the pro-p topology on each of the factors.