• 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 the Korean Mathematical Society
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• v.46 no.2
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• pp.249-256
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• 2009

Let $G{\otimes}G$ be the tensor square of a group G. The set of all elements a in G such that $a{\otimes}g\;=\;1_{\otimes}$, for all g in G, is called the tensor centre of G and denoted by $Z^{\otimes}^$(G). In this paper some properties of the tensor centre of G are obtained and the capability of the pair of groups (G, G') is determined. Finally, the structure of $J_2$(G) will be described, where $J_2$(G) is the kernel of the map $\kappa$ : $G{\otimes}\;{\rightarrow}\;G'$.

• 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 $\mathbb{Z}_2{\times}\mathbb{Z}_2$.

• Bulletin of the Korean Mathematical Society
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• v.56 no.4
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• pp.815-827
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• 2019

It is well known that the denominator of the Dedekind sum s(c, d) divides 2 gcd(d, 3)d and that no smaller denominator independent of c can be expected. In contrast, here we prove that we usually get a smaller denominator in S(H, d), the sum of the s(c, d)'s over all the c's in a subgroup H of order n > 1 in the multiplicative group $(\mathbb{Z}/d\mathbb{Z})^*$. First, we prove that for p > 3 a prime, the sum 2S(H, p) is a rational integer of the same parity as (p-1)/2. We give an application of this result to upper bounds on relative class numbers of imaginary abelian number fields of prime conductor. Finally, we give a general result on the denominator of S(H, d) for non necessarily prime d's. We show that its denominator is a divisor of some explicit divisor of 2d gcd(d, 3).