Groups St Andrews 2001 in Oxford: On distances of 2-groups and 3-groups
Groups St Andrews 2001 in Oxford: On distances of 2-groups and 3-groups
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2001 年牛津圣安德鲁斯分组赛：关于 2 组和 3 组的距离
DOI：
10.1017/cbo9780511542770.018
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发表时间：
2003
期刊：
影响因子：
--
通讯作者：
A. Drápal
中科院分区：
文献类型：
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作者：
A. Drápal
本文研究n阶有限群G（n）和G（n）不同构，且其中{（u，v）∈ G×G; u ∈ v 6= u ∈ v}的大小是最小可能的（关于给定的n）.它考察了2-群的情况，讨论了已知结果到p-群的可能推广，p是奇素数，并建立了当G（G）是初等交换3-群时的最小可能距离。设G（n）和G（n）是n阶有限群.考虑集合{（u，v）∈ G × G; u <$v 6= u <$v}，用d（u，v）表示它的大小。数d（，）称为和的（汉明）距离。如果d（n，n）<n ~ 2/4，n是2的幂，则G（n）= G（n），由[4]。第1节列出了关于2-群的距离的进一步结果，而第2节讨论了相关的证明机制，以及它可能推广到p-群，p是奇素数。在第二节中，我们也给出了非同构p-群G（n）和G（n），|G| = n > p，其中d（n，n）= n2（p2 − 1）/（4p 2）。当G（n）是初等交换3-群时，这个结果是可能的最好结果--在第三节中我们将证明，在这种情况下，d（n，n）<2n ~ 2/9意味着G（n）n = G（n）。若H ≤ G（H），则G（H）中H的所有左（或右）陪集的集合分别记为L（H）和R（H）。若A ∈ G，B ∈ G，则{（u，v）∈ A×B; u ∈ v 6= u ∈ v}的大小记为d（A，B）.
This paper is concerned with finite groups G(◦) and G(∗) of order n that are not isomorphic, and where the size of {(u, v) ∈ G×G; u◦v 6= u∗v} is the least possible (with respect to the given n). It surveys the case of 2-groups, discusses the possible generalization of the known results to p-groups, p an odd prime, and establishes the least possible distance in the case when G(∗) is an elementary abelian 3-group. Let G(◦) and G(∗) be finite groups of order n. Consider the set {(u, v) ∈ G × G; u ◦ v 6= u ∗ v} and denote its size by d(◦, ∗). The number d(◦, ∗) is called the (Hamming) distance of ◦ and ∗. If d(◦, ∗) < n2/4 and n is a power of two, then G(◦) ∼= G(∗), by [4]. Section 1 lists further results about distances of 2groups, while Section 2 discusses the associated proof machinery, and its possible generalization to p-groups, p an odd prime. In Section 2 there are also presented non-isomorphic p-groups G(◦) and G(∗), |G| = n > p, for which d(◦, ∗) = n2(p2 − 1)/(4p2). This result is the best possible, when G(◦) is an elementary abelian 3-group—in Section 3 we shall show that in such a case d(◦, ∗) < 2n2/9 implies G(◦) ∼= G(∗). If H ≤ G(◦), then the set of all left (or right) cosets of H in G(◦) is denoted by L(H) and R(H), respectively. If A ⊆ G and B ⊆ G, then the size of {(u, v) ∈ A×B; u ◦ v 6= u ∗ v} will be denoted by d(A,B).