• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.1031-1038
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• 2021

The main aim of this article is to study quantitative structure of finite simple exceptional groups F₄(2ⁿ) with n > 1. Here, we prove that the finite simple exceptional groups F₄(2ⁿ), where 2⁴ⁿ + 1 is a prime number with n > 1 a power of 2, can be uniquely determined by their orders and the set of the number of elements with the same order. In conclusion, we give a positive answer to J. G. Thompson's problem for finite simple exceptional groups F₄(2ⁿ).

• Bulletin of the Korean Mathematical Society
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• v.43 no.2
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• pp.353-375
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• 2006

In this paper we establish a counting method for the Green classes of the Birget-rhodes expansion of finite groups. As an application of the results, we derive explicit enumeration formulas for the Green classes for finite groups of order pq and a finite cyclic group of order $p^m$, where p and q are arbitrary given distinct prime numbers.

• Korean Journal of Mathematics
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• v.32 no.1
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• pp.1-14
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• 2024

We seek for an invertible map B from L²(Γ) to L²(G), where G is a finite abelian group and Γ is the direct product of finite cyclic groups which is isomorphic to G, so that any Gabor frame in L²(G), is a generalized pseudo B-Gabor frame.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1195-1204
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• 2010

In this paper, we prove a criterion of conjugacy separability of generalized free products of polycyclic-by-finite groups with a non cyclic amalgamated subgroup. Applying this criterion, we prove that certain generalized free products of polycyclic-by-finite groups are conjugacy separable.

• Journal of the Korean Mathematical Society
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• v.42 no.4
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• pp.795-826
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• 2005

We study free actions of finite abelian groups on 3­dimensional nilmanifolds. 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. All such actions are completely classified.

• Communications of the Korean Mathematical Society
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• v.21 no.1
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• pp.17-25
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• 2006

In this paper, we study the right regular representation $R_\Gamma$ of a finite group G on the vector space consisting of vector valued functions on ${\Gamma}/G$ with a subgroup r of G and give a trace formula using the work of M. -F. Vigneras.

• Bulletin of the Korean Mathematical Society
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• v.50 no.2
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• pp.697-704
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• 2013

The function $g{\mapsto}{\zeta}^k_G(g)$ which counts the number of solutions of $x^k=g$ in a finite group G, is not necessarily a character of G. We study this function for the case of dihedral groups and generalized quaternion groups.

• 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.

• East Asian mathematical journal
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• v.30 no.3
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• pp.293-302
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• 2014

In this paper we provide a rigorous proof for the fact that there are exactly 8 connected Alexander quandles of order $2^5$ by combining properties of fixed point free automorphisms of finite abelian 2-groups and the classification of conjugacy classes of GL(5, 2). Furthermore we verify that six of the eight associated Alexander modules are simple, whereas the other two are semisimple.

• Bulletin of the Korean Mathematical Society
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• v.53 no.2
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• pp.479-486
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• 2016

The norm N(G) of a group G is the intersection of the normalizers of all the subgroups of G. In this paper, the structure of finite groups with a cyclic norm quotient is determined. As an application of the result, an interesting characteristic of cyclic groups is given, which asserts that a finite group G is cyclic if and only if Aut(G)/P(G) is cyclic, where P(G) is the power automorphism group of G.