• 대한수학회보
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• 제49권5호
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• pp.949-959
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• 2012

In general, a class-preserving automorphism of generalized free products of nilpotent groups, amalgamating a cyclic normal subgroup of order 8, need not be an inner automorphism. We prove that every class-preserving automorphism of generalized free products of nitely generated nilpotent groups, amalgamating a cyclic normal subgroup of order less than 8, is inner.

• 대한수학회지
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• 제50권4호
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• pp.813-828
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• 2013

It is known that generalized free products of finitely generated nilpotent groups are conjugacy separable when the amalgamated subgroups are cyclic or central in both factor groups. However, those generalized free products may not be conjugacy separable when the amalgamated subgroup is a direct product of two infinite cycles. In this paper we show that generalized free products of finitely generated nilpotent groups are conjugacy separable when the amalgamated subgroup is ${\langle}h{\rangle}{\times}D$, where D is in the center of both factors.

• 대한수학회보
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• 제48권6호
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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.

• 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.

• 대한수학회지
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• 제40권2호
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• pp.225-239
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• 2003

(G, <) is an ordered group if'<'is a total order relation on G in which f < g implies that xfy < xgy for all f, g, x, y $\in$ G. We say that (G, <) is centrally ordered if (G, <) is ordered and ［G,D］ $\subseteq$ C for every convex jump C $\prec$ D in G. Equivalently, if $f^{-1}g f{\leq} g^2$ for all f, g $\in$ G with g > 1. Every order on a torsion-free locally nilpotent group is central. We prove that if every order on every two-generator subgroup of a locally soluble orderable group G is central, then G is locally nilpotent. We also provide an example of a non-nilpotent two-generator metabelian orderable group in which all orders are central.

• 대한수학회보
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• 제60권5호
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• pp.1375-1390
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• 2023

In this note, we shall show that the generalized free products of subgroup separable groups amalgamating a subgroup which itself is a finite extension of a finitely generated normal subgroup of both the factor groups are weakly potent and cyclic subgroup separable. Then we apply our result to generalized free products of finite extensions of finitely generated torsion-free nilpotent groups. Finally, we shall show that their tree products are cyclic subgroup separable.

• 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.

• 대한수학회보
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• 제47권6호
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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.

• 대한수학회지
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• 제32권2호
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• pp.289-300
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• 1995

Every group G has a unique maximal normal locally nilpotent subgroup $\Phi(G)$, called the Hirsh-Plotkin radical of G [9]. If G is a group, we define the upper Hirsh-Plotkin series of G to be the ascending series $1 = R_0 \leq R_1 \leq \ldots$ in which $R_{\alpha+1}/R_\alpha = \{Phi(G/R_\alpha)$ for each ordinal $\alpha and R_\beta = \cup_{\alpha<\beta}R_\alpha$ for each limit ordinal $\beta$. If $R_r = G$ for some natural number r, then G is said to have locally nilpotent length r. $(LN)^r$ denotes the calss of groups of locally nilpotent length at most r.

• Kyungpook Mathematical Journal
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• 제63권2호
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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.