• Korean Journal of Mathematics
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• v.8 no.2
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• pp.173-180
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• 2000

Let $k$ be a real abelian field such that the conductor of every nontrivial character belonging to $k$ agrees with the conductor of $k$. Note that real quadratic fields satisfy this condition. For a prime $p$, let $k_{\infty}$ be the $\mathbb{Z}_p$-extension of $k$. The aim of this paper is to produce a set of generators of the Tate cohomology group $\hat{H}^{-1}$ of the circular units of $k_n$, the $nth$ layer of the $\mathbb{Z}_p$-extension of $k$, where $p$ is an odd prime. This result generalizes some earlier works which treated the case when $k$ is real quadratic field and used them to study ${\lambda}$-invariants of $k$.

• Korean Journal of Mathematics
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• v.4 no.1
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• pp.45-49
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• 1996

Let $F_0$ be the maximal real subfield of $\mathbb{Q}({\zeta}_q+{\zeta}_q^{-1})$ and $F_{\infty}={\cup}_{n{\geq}0}F_n$ be its basic $\mathbb{Z}_p$-extension. Let $A_n$ be the Sylow $p$-subgroup of the ideal class group of $F_n$. The aim of this paper is to examine the injectivity of the natural $mapA_n{\rightarrow}A_m$ induced by the inclusion $F_n{\rightarrow}F_m$ when $m>n{\geq}0$. By using cyclotomic units of $F_n$ and by applying cohomology theory, one gets the following result: If $p$ does not divide the order of $A_1$, then $A_n{\rightarrow}A_m$ is injective for all $m>n{\geq}0$.

• Korean Journal of Mathematics
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• v.5 no.2
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• pp.185-193
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• 1997

In this paper, we study the $BP_*$-module structure of $BP_*$(BG) mod $(p,v_1,{\cdots})^2$ for non abelian groups of the order $p^3$. We know $grBP_*(BG)=BP_*{\otimes}H(H_*(BG);Q_1){\oplus}BP^*/(p,v_1){\otimes}ImQ_1$. The similar fact occurs for $BP_*$-homology $grBP_*(BG)=BP_*s^{-1}H(H_*(BG);Q_1){\oplus}BP_*/(p,v)s^{-1}H^{odd}(BG)$ by using the spectral sequence $E^{*,*}_2=Ext_{BP^*}(BP_*(BG),BP^*){\Rightarrow}BP^*(BG)$.

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.75-96
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• 2006

Let m, n be positive integers. We denote by R(m,n) (respectively P(m,n)) the class of all groups G such that, for every n subsets $X_1,X_2\ldots,X_n$, of size m of G there exits a non-identity permutation $\sigma$ such that $X_1X_2{\cdots}X_n{\cap}X_{\sigma(1)}X_{/sigma(2)}{\cdots}X_{/sigma(n)}\neq\phi$ (respectively $X_1X_2{\cdots}X_n=X_{/sigma(1)}X_{\sigma(2)}{\cdots}X_{\sigma(n)}$). Let G be a non-abelian group. In this paper we prove that (i) $G{\in}P$(2,3) if and only if G isomorphic to $S_3$, where $S_n$ is the symmetric group on n letters. (ii) $G{\in}R$(2, 2) if and only if ${\mid}G{\mid}\geq8$. (iii) If G is finite, then $G{\in}R$(3, 2) if and only if ${\mid}G{\mid}\geq14$ or G is isomorphic to one of the following: SmallGroup(16, i), $i\in$ {3, 4, 6, 11, 12, 13}, SmallGroup(32, 49), SmallGroup(32, 50), where SmallGroup(m, n) is the nth group of order m in the GAP [13] library.

• Kyungpook Mathematical Journal
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• v.64 no.2
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• pp.303-309
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• 2024

An imaginary bicyclic biquadratic number field K is a field of the form ${\mathbb{Q}}({\sqrt{-m}},{\sqrt{-n}})$ where m and n are squarefree positive integers. The ideal class number h_K of K is the order of the abelian group I_K/P_K, where I_K and P_K are the groups of fractional and principal fractional ideals in the ring of integers 𝒪_K of K, respectively. This provides a measure on how far is 𝒪_K from being a PID. We determine all imaginary bicyclic biquadratic number fields with class number 5. We show there are exactly 243 such fields.

• Communications of the Korean Mathematical Society
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• v.28 no.4
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• pp.643-647
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• 2013

Let G be a group and let $p$ be a prime number. If the set $\mathcal{A}(G)$ of all commuting automorphisms of G forms a subgroup of Aut(G), then G is called $\mathcal{A}(G)$-group. In this paper we show that any $p$-group with cyclic maximal subgroup is an $\mathcal{A}(G)$-group. We also find the structure of the group $\mathcal{A}(G)$ and we show that $\mathcal{A}(G)=Aut_c(G)$. Moreover, we prove that for any prime $p$ and all integers $n{\geq}3$, there exists a non-abelian $\mathcal{A}(G)$-group of order $p^n$ in which $\mathcal{A}(G)=Aut_c(G)$. If $p$ > 2, then $\mathcal{A}(G)={\cong}\mathbb{Z}_p{\times}\mathbb{Z}_{p^{n-2}}$ and if $p=2$, then $\mathcal{A}(G)={\cong}\mathbb{Z}_2{\times}\mathbb{Z}_2{\times}\mathbb{Z}_{2^{n-3}}$ or $\mathbb{Z}_2{\times}\mathbb{Z}_2$.

• Communications of Mathematical Education
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• v.5
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• pp.523-523
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• 1997

Suppose that G is a group of permutations of a set ${\Omega}$. For a finite subset ${\gamma}$of${\Omega}$, the movement of ${\gamma}$ under the action of G is defined as move(${\gamma}$):=$max\limits_{g{\epsilon}G}|{\Gamma}^{g}{\backslash}{\Gamma}|$, and ${\gamma}$ will be said to have restricted movement if move(${\gamma}$)<|${\gamma}$|. Moreover if, for an infinite subset ${\gamma}$of${\Omega}$, the sets|{\Gamma}^{g}{\backslash}{\Gamma}| are finite and bounded as g runs over all elements of G, then we may define move(${\gamma}$)in the same way as for finite subsets. If move(${\gamma}$)${\leq}$m for all ${\gamma}$${\subseteq}$${\Omega}$, then G is said to have bounded movement and the movement of G move(G) is defined as the maximum of move(${\gamma}$) over all subsets ${\gamma}$ of ${\Omega}$. Having bounded movement is a very strong restriction on a group, but it is natural to ask just which permutation groups have bounded movement m. If move(G)=m then clearly we may assume that G has no fixed points is${\Omega}$, and with this assumption it was shown in [4, Theorem 1]that the number t of G=orbits is at most 2m-1, each G-orbit has length at most 3m, and moreover|${\Omega}$|${\leq}$3m+t-1${\leq}$5m-2. Moreover it has recently been shown by P. S. Kim, J. R. Cho and C. E. Praeger in [1] that essentially the only examples with as many as 2m-1 orbits are elementary abelian 2-groups, and by A. Gardiner, A. Mann and C. E. Praeger in [2,3]that essentially the only transitive examples in a set of maximal size, namely 3m, are groups of exponent 3. (The only exceptions to these general statements occur for small values of m and are known explicitly.) Motivated by these results, we would decide what role if any is played by primes other that 2 and 3 for describing the structure of groups of bounded movement.