• Journal of the Korean Mathematical Society
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• v.54 no.5
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• pp.1411-1440
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• 2017

We study free actions of finite groups on 3-dimensional nil-manifolds with the first homology ${\mathbb{Z}}^2{\oplus}{\mathbb{Z}}_p$. By the works of Bieberbach and Waldhausen, this classification problem is reduced to classifying all normal nilpotent subgroups of almost Bieberbach groups of finite index, up to affine conjugacy.

• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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• pp.109-114
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• 1989

This paper deals with the classifying spaces of finite groups. To any finite group G we associate a space BG with the property that .pi.$_{1}$(BG)=G, .pi.$_{i}$ (BG)=0 for i>1. BG is called the classifying space of G. Consider the problem of finding a stable splitting BG= $X_{1}$$^{V}$ $X_{1}$$^{V}$..$^{V}$ $X_{n}$ localized at pp. Ideally the $X_{i}$ 's are indecomposable, thus displaying the homotopy type of BG in the simplest terms. Such a decomposition naturally splits $H^{*}$(BG). The main purpose of this paper is to give the classification theorem in stable homotopy theory for groups with periodic cohomology i.e. cyclic Sylow p-subgroups for p an odd prime and to calculate some universal stable element. In this paper, all cohomology groups are with Z/p-coefficients and p is an odd prime.prime.

• Kyungpook Mathematical Journal
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• v.63 no.2
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• pp.167-173
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• 2023

Let G be a finite group and let ν(G) be the probability that two randomly selected elements of G produce a nilpotent group. In this article we show that for every positive integer n > 0, there is a finite group G such that ${\nu}(G)={\frac{1}{n}}$. We also classify all groups G with ${\nu}(G)={\frac{1}{2}}$. Further, we prove that if G is a solvable nonnilpotent group of even order, then ${\nu}(G){\leq}{\frac{p+3}{4p}}$, where p is the smallest odd prime divisor of |G|, and that equality exists if and only if $\frac{G}{Z_{\infty}(G)}$ is isomorphic to the dihedral group of order 2p where Z_∞(G) is the hypercenter of G. Finally we find an upper bound for ν(G) in terms of |G| where G ranges over all groups of odd square-free order.

• Bulletin of the Korean Mathematical Society
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• v.38 no.4
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• pp.719-725
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• 2001

For a p-compact group G, if G-space X is of finite S-type then we show that the localization $S^{-1}H^*(X_{hG})$is zero. By using this result, we prove the localization theorem for the pair G-space (X, A)

• Journal of applied mathematics & informatics
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• v.27 no.3_4
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• pp.653-659
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• 2009

It is proved that if $M\;=\;B_p(3)$ or $C_p(3)$, p is an odd prime, G is a finite group and has the same order components of M, then $G\;{\cong}\;Bp(3)$ or $C_p(3)$.

• Journal of the Chungcheong Mathematical Society
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• v.33 no.1
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• pp.113-132
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• 2020

We show that if a finite group G acts freely with homotopically trivial translations on a 3-dimensional nilmanifold 𝓝_p with the first homology ℤ² ⊕ ℤ_p, then either G is cyclic or there exist finite nonabelian groups acting freely on 𝓝_p which yield orbit manifolds homeomorphic to 𝓝/𝜋₃ or 𝓝/𝜋₄.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.751-763
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• 2016

The properties of the nonabelian tensor products are interesting in different contexts of algebraic topology and group theory. We prove two theorems, dealing with the nonabelian tensor products of projective limits of finite groups. The first describes their topology. Then we show a result of embedding in the second homology group of a pro-p-group, via the notion of complete exterior centralizer. We end with some open questions, originating from these two results.

• Journal of the Korean Mathematical Society
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• v.45 no.1
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• pp.137-150
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• 2008

In this paper we will prove that the simple groups $B_p(3)\;and\;G_p(3)$, p an odd prime number, are 2-recognizable by the set of their order components. More precisely we will prove that if G is a finite group and OC(G) denotes the set of order components of G, then OC(G) = $OC(B_p(3))$ if and only if $G{\cong}B_p(3)\;or\;C_p(3)$.

• Honam Mathematical Journal
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• v.36 no.2
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• pp.417-424
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• 2014

Let A be an abelian variety defined over a number field K and p be a prime. Define ${\varphi}_i=(x^{p^i}-1)/(x^{p^{i-1}}-1)$. Let $A_{{\varphi}i}$ be the abelian variety defined over K associated to the polynomial ${\varphi}i$ and let Ш($A_{{\varphi}i}$) denote the Tate-Shafarevich groups of $A_{{\varphi}i}$ over K. In this paper assuming Ш(A/F) is finite, we compute [Ш($A_{{\varphi}1}$)][Ш($A_{{\varphi}2}$)]/[Ш($A_{{\varphi}1{\varphi}2}$)] in terms of K-rational points of $A_{{\varphi}i}$, $A_{{\varphi}1{\varphi}2}$ and their dual varieties, where [X] is the order of a finite abelian group X.

• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.781-787
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• 2022

In this paper, we show that under certain conditions the Wedderburn decomposition of a finite semisimple group algebra 𝔽_qG can be deduced from a subalgebra 𝔽_q(G/H) of factor group G/H of G, where H is a normal subgroup of G of prime order P. Here, we assume that q = p^r for some prime p and the center of each Wedderburn component of 𝔽_qG is the coefficient field 𝔽_q.