• Communications of the Korean Mathematical Society
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• v.28 no.1
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• pp.55-62
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• 2013

In [5], Johnson introduced the primitivity of subgroups and proved that a finite group G is supersolvable if every primitive subgroup of G has a prime power index in G. In that paper, he also posed an interesting problem: what a group looks like if all of its primitive subgroups are maximal. In this note, we give the detail structure of such groups in solvable case. Finally, we use the primitivity of some subgroups to characterize T-group and the solvable $PST_0$-groups.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.719-726
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• 1998

In this paper, we consider conjugacy of subgroups of some split metacyclic groups of odd prime power order to determine the numbers of conjugacy classes of subgroups of those groups. The study was motivated by the linear isomorphism problem of metacyclic primitive linear groups.

• Bulletin of the Korean Mathematical Society
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• v.59 no.6
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• pp.1511-1522
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• 2022

Let H be a subgroup of $\mathbb{Z}^*_n$ (the multiplicative group of integers modulo n) and h₁, h₂, …, h_l distinct representatives of the cosets of H in $\mathbb{Z}^*_n$. We now define a polynomial J_n,H(x) to be $$J_{n,H}(x)=\prod^l_{j=1} \left( x-\sum_{h{\in}H} {\zeta}^{h_jh}_n\right)$$, where ${\zeta}_n=e^{\frac{2{\pi}i}{n}}$ is the nth primitive root of unity. Polynomials of such form generalize the nth cyclotomic polynomial $\Phi_n(x)={\prod}_{k{\in}\mathbb{Z}^*_n}(x-{\zeta}^k_n)$ as J_n,{1}(x) = Φ_n(x). While the nth cyclotomic polynomial Φ_n(x) is irreducible over ℚ, J_n,H(x) is not necessarily irreducible. In this paper, we determine the subgroups H for which J_n,H(x) is irreducible over ℚ.

• KIPS Transactions on Software and Data Engineering
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• v.9 no.2
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• pp.39-44
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• 2020

In this paper, we show a modified Diffie-Hellman key exchange protocol that can exchange keys by exposing only minimal information using pre-computable session key pairs. The discrete logarithm problem, which provides the safety of existing Diffie-Hellman and Diffie-Hellman based techniques, is modified to prevent exposure of primitive root. We prove the algorithm's operation by applying the actual value to the proposed scheme and compare the execution time and safety with the existing algorithm, shown that the security of the algorithm is improved more than the product of the time complexity of the two base algorithms while maintaining the computation amount at the time of key exchange. Based on the proposed algorithm, it is expected to provide a key exchange environment with improved security.

• Animal Systematics, Evolution and Diversity
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• v.18 no.1
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• pp.165-179
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• 2002

In spite of dramatic change of environmental condition since Cambrian big-bang (explosion occurred ca.540 mya, the phylum Arthropoda retains a great diversity, and it is estimated approximately that 1-10 million arthropod species are extant on the earth. Except for an extinct arthopod subphylum Trilobita, extant arthropods could be divided into five subphyla: Hexapoda, Crustacea, Myriapoda, Chelicerata, and Pycnosonida. During the last century, systematists have disputed about interrealtionships among Arthropoda and its relatives (Onychophora, Tardigrada, and Pentastomida), arthropod phylogenetic position within protostome animals, monophyly or polyphyly of the phylum Arthropods, and interrelationships among five arthropod subgroups (subphyla) etc. Recently, new animal phylogeny was reported that protostomes could be clustered into two groups, Lophotrochozoa and Ecdysozoa, and molting animals such as Nematoda and Arthropoda were included within the Ecdysozoa. On the basis of the new animal phylogeny, first of all, I would mention phylogenetic positions and relationships of Arthropods and its relatives to introduce controversies of arthropod phylogeny in phylum level of animals. After that, I focused mainly on the controveries related to arthropod monophyly and phylogenetic relationships among four major arthropod groups except Pycnogonida. In this work, Pycnogonida which is a relatively small group and one of the five arthropod subphyla was not handled significantly although there are some controversies if it is a sister taxon of chelicerates or the most primitive arthropod group (namely, a sister of four remains arthropod groups).