• Title, Summary, Keyword: finite group

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THE STRUCTURE OF SEMIPERFECT RINGS

  • Han, Jun-Cheol
    • Journal of the Korean Mathematical Society
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    • v.45 no.2
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    • pp.425-433
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    • 2008
  • Let R be a ring with identity $1_R$ and let U(R) denote the group of all units of R. A ring R is called locally finite if every finite subset in it generates a finite semi group multiplicatively. In this paper, some results are obtained as follows: (1) for any semilocal (hence semiperfect) ring R, U(R) is a finite (resp. locally finite) group if and only if R is a finite (resp. locally finite) ring; U(R) is a locally finite group if and only if U$(M_n(R))$ is a locally finite group where $M_n(R)$ is the full matrix ring of $n{\times}n$ matrices over R for any positive integer n; in addition, if $2=1_R+1_R$ is a unit in R, then U(R) is an abelian group if and only if R is a commutative ring; (2) for any semiperfect ring R, if E(R), the set of all idempotents in R, is commuting, then $R/J\cong\oplus_{i=1}^mD_i$ where each $D_i$ is a division ring for some positive integer m and |E(R)|=$2^m$; in addition, if 2=$1_R+1_R$ is a unit in R, then every idempotent is central.

IRREDUCIBLE REPRESENTATIONS OF SOME METACYCLIC GROUPS WITH AN APPLICATION

  • Sim, Hyo-Seob
    • East Asian mathematical journal
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    • v.33 no.1
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    • pp.45-52
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    • 2017
  • Motivated by the problem of determining all right ideals of a group algebra FG for a finite group G over a finite field F, we explicitly determine the faithful irreducible representations of some finite metacylic groups over finite fields. By using that result, we determine the structure of all right ideals of the group algebra for the symmetric group $S_3$ over a finite field F, as an example.

ON DECOMPOSABILITY OF FINITE GROUPS

  • Arhrafi, Ali-Reza
    • Journal of the Korean Mathematical Society
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    • v.41 no.3
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    • pp.479-487
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    • 2004
  • Let G be a finite group and N be a normal subgroup of G. We denote by ncc(N) the number of conjugacy classes of N in G and N is called n-decomposable, if ncc(N) = n. Set $K_{G}\;=\;\{ncc(N)$\mid$N{\lhd}G\}$. Let X be a non-empty subset of positive integers. A group G is called X-decomposable, if KG = X. In this paper we characterise the {1, 3, 4}-decomposable finite non-perfect groups. We prove that such a group is isomorphic to Small Group (36, 9), the $9^{th}$ group of order 36 in the small group library of GAP, a metabelian group of order $2^n{2{\frac{n-1}{2}}\;-\;1)$, in which n is odd positive integer and $2{\frac{n-1}{2}}\;-\;1$ is a Mersenne prime or a metabelian group of order $2^n(2{\frac{n}{3}}\;-\;1)$, where 3$\mid$n and $2\frac{n}{3}\;-\;1$ is a Mersenne prime. Moreover, we calculate the set $K_{G}$, for some finite group G.

FINITE EXTENSIONS OF WEIGHTED WORD L-DELTA GROUPS

  • Ryang, Do-Hyoung
    • The Pure and Applied Mathematics
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    • v.15 no.4
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    • pp.353-364
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    • 2008
  • The purpose of this paper is to investigate the finite extension of weighted word L-delta groups. The paper revealed that a finite extension of a weighted word L-delta group is a weighted word L-delta group, and an abelian group, in addition, is a weighted word L-delta group and simultaneously a word L-delta group.

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POLYGONAL PRODUCTS OF RESIDUALLY FINITE GROUPS

  • Wong, Kok-Bin;Wong, Peng-Choon
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.1
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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.

HOMOTOPY FIXED POINT SET $FOR \rho-COMPACT$ TORAL GROUP

  • Lee, Hyang-Sook
    • Bulletin of the Korean Mathematical Society
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    • v.38 no.1
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    • pp.143-148
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    • 2001
  • First, we show the finiteness property of the homotopy fixed point set of p-discrete toral group. Let $G_\infty$ be a p-discrete toral group and X be a finite complex with an action of $G_\infty such that X^K$ is nilpotent for each finit p-subgroup K of $G_\infty$. Assume X is $F_\rho-complete$. Then X(sup)hG$\infty$ is F(sub)p-finite. Using this result, we give the condition so that X$^{hG}$ is $F_\rho-finite for \rho-compact$ toral group G.

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FINITE GROUP ACTIONS ON THE 3-DIMENSIONAL NILMANIFOLD

  • Goo, Daehwan;Shin, Joonkook
    • Journal of the Chungcheong Mathematical Society
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    • v.18 no.2
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    • pp.223-232
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    • 2005
  • We study only free actions of finite groups G on the 3-dimensional nilmanifold, up to topological conjugacy which yields an infra-nilmanifold of type 2.

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