• 대한수학회지
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• 제34권1호
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• pp.227-235
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• 1997

In recent years there has been much interest in the study of groups satisfying various permutability conditions (see, for instance, [1], [2] and [3]). More recently, the following condition has been studied: for some , if S is any subset of m elements of a group G, then $$\mid$S^2$\mid$ < m^2$ (where, for subsets A, B of G, AB stands for ${ab; a \in A, b \in B}$).

• 대한수학회보
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• 제58권1호
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• pp.225-234
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• 2021

Let D be a division ring and n > 1 be an integer. In this paper, it is shown that if D ≠ ��3, then every subpermutable subgroup of the general skew linear group GLn(D) is normal. By applying this result, we show that every subpermutable subgroup of the unit group (ℝG)∗ of the real group algebras RG of finite groups G is normal in (ℝG)∗.

• ETRI Journal
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• 제36권1호
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• pp.124-133
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• 2014

To speed up data-intensive programs, two complementary techniques, namely nested loops parallelization and data locality optimization, should be considered. Effective parallelization techniques distribute the computation and necessary data across different processors, whereas data locality places data on the same processor. Therefore, locality and parallelization may demand different loop transformations. As such, an integrated approach that combines these two can generate much better results than each individual approach. This paper proposes a unified approach that integrates these two techniques to obtain an appropriate loop transformation. Applying this transformation results in coarse grain parallelism through exploiting the largest possible groups of outer permutable loops in addition to data locality through dependence satisfaction at inner loops. These groups can be further tiled to improve data locality through exploiting data reuse in multiple dimensions.

• Journal of applied mathematics & informatics
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• 제20권1_2호
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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.