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首页> 外文期刊>Information Theory, IEEE Transactions on >Finite p-Groups, Entropy Vectors, and the Ingleton Inequality for Nilpotent Groups
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Finite p-Groups, Entropy Vectors, and the Ingleton Inequality for Nilpotent Groups

机译：有限p群，熵向量和幂等群的Ingleton不等式

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摘要

In this paper, we study the capacity/entropy region of finite, directed, acyclic, multiple-sources, and multiple-sinks network by means of group theory and entropy vectors coming from groups. There is a one-to-one correspondence between the entropy vector of a collection of (n) random variables and a certain group-characterizable vector obtained from a finite group and (n) of its subgroups. We are looking at nilpotent group characterizable entropy vectors and show that they are all also abelian group characterizable, and hence they satisfy the Ingleton inequality. It is known that not all entropic vectors can be obtained from abelian groups, so our result implies that to get more exotic entropic vectors, one has to go at least to soluble groups or larger nilpotency classes. The result also implies that Ingleton inequality is satisfied by nilpotent groups of bounded class, depending on the order of the group.
机译：在本文中，我们利用群论和来自群的熵向量研究有限，有向，无环，多源和多宿网络的容量/熵区域。 （n）个随机变量的集合的熵向量与从有限组及其（n）个子组获得的某些可表征组的向量之间存在一一对应的关系。我们正在研究幂等群可表征的熵向量，并证明它们都是阿贝尔群可表征的，因此它们满足Ingleton不等式。众所周知，并非所有熵矢量都可以从阿贝尔群中获得，因此我们的结果表明，要获得更多奇特的熵矢量，必须至少进入可溶性基团或更大的幂等性类。该结果还暗示，有界类的幂等组可以满足Ingleton不等式，具体取决于该组的顺序。

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