On Abelian group representability of finite groups
Thomas, Eldho K.
Date of Issue2014
School of Physical and Mathematical Sciences
A set of quasi-uniform random variables X1,…,Xn may be generated from a finite group G and n of its subgroups, with the corresponding entropic vector depending on the subgroup structure of G. It is known that the set of entropic vectors obtained by considering arbitrary finite groups is much richer than the one provided just by abelian groups. In this paper, we start to investigate in more detail different families of non-abelian groups with respect to the entropic vectors they yield. In particular, we address the question of whether a given non-abelian group G and some fixed subgroups G1,…,Gn end up giving the same entropic vector as some abelian group A with subgroups A1,…,An, in which case we say that (A,A1,…,An) represents (G,G1,…,Gn). If for any choice of subgroups G1,…,Gn, there exists some abelian group A which represents G, we refer to G as being abelian (group) representable for n. We completely characterize dihedral, quasi-dihedral and dicyclic groups with respect to their abelian representability, as well as the case when n=2, for which we show a group is abelian representable if and only if it is nilpotent. This problem is motivated by understanding non-linear coding strategies for network coding, and network information theory capacity regions.
Advances in mathematics of communications
© 2014 AIMS. This paper was published in Advances in Mathematics of Communications and is made available as an electronic reprint (preprint) with permission of AIMS. The paper can be found at the following official DOI: [http://dx.doi.org/10.3934/amc.2014.8.139]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.