# American Institute of Mathematical Sciences

May  2014, 8(2): 139-152. doi: 10.3934/amc.2014.8.139

## On Abelian group representability of finite groups

 1 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

Received  December 2012 Published  May 2014

A set of quasi-uniform random variables $X_1,\ldots,X_n$ 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 $G_1,\ldots,G_n$ end up giving the same entropic vector as some abelian group $A$ with subgroups $A_1,\ldots,A_n$, in which case we say that $(A, A_1, \ldots, A_n)$ represents $(G, G_1, \ldots, G_n)$. If for any choice of subgroups $G_1,\ldots,G_n$, 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.
Citation: Eldho K. Thomas, Nadya Markin, Frédérique Oggier. On Abelian group representability of finite groups. Advances in Mathematics of Communications, 2014, 8 (2) : 139-152. doi: 10.3934/amc.2014.8.139
##### References:
 [1] T. H. Chan, Aspects of Information Inequalities and its Applications, M.Phil Thesis, Dept. of Information Engineering, The Chinese University of Hong Kong, 1998. Google Scholar [2] T. H. Chan, Group characterizable entropy functions, in 2007 IEEE International Symposium on Information Theory, Nice, France, 2007. doi: 10.1109/ISIT.2007.4557275.  Google Scholar [3] T. H. Chan and R. W. Yeung, On a relation between information inequalities and group theory, IEEE Trans. on Information Theory, 48 (2002), 1992-1995. doi: 10.1109/TIT.2002.1013138.  Google Scholar [4] D. S. Dummit and R. M. Foote, Abstract Algebra, Third edition, John Wiley and Sons, 2004.  Google Scholar [5] B. Hassibi and S. Shadbakht, Normalized entropy vectors, network information theory and convex optimization, in 2007 Information Theory Workshop, 2007. doi: 10.1109/ITWITWN.2007.4318051.  Google Scholar [6] E. Thomas and F. Oggier, A note on quasi-uniform distributions and abelian group representability, in 2012 International Conference on Signal Processing and Communications, Bangalore, India, 2012. doi: 10.1109/SPCOM.2012.6290020.  Google Scholar [7] X. Yan, R. Yeung and Z. Zhang, The capacity for multi-source multi-sink network coding, in 2007 International Symposium on Information Theory, Nice, France, 2007. doi: 10.1109/ISIT.2007.4557213.  Google Scholar

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##### References:
 [1] T. H. Chan, Aspects of Information Inequalities and its Applications, M.Phil Thesis, Dept. of Information Engineering, The Chinese University of Hong Kong, 1998. Google Scholar [2] T. H. Chan, Group characterizable entropy functions, in 2007 IEEE International Symposium on Information Theory, Nice, France, 2007. doi: 10.1109/ISIT.2007.4557275.  Google Scholar [3] T. H. Chan and R. W. Yeung, On a relation between information inequalities and group theory, IEEE Trans. on Information Theory, 48 (2002), 1992-1995. doi: 10.1109/TIT.2002.1013138.  Google Scholar [4] D. S. Dummit and R. M. Foote, Abstract Algebra, Third edition, John Wiley and Sons, 2004.  Google Scholar [5] B. Hassibi and S. Shadbakht, Normalized entropy vectors, network information theory and convex optimization, in 2007 Information Theory Workshop, 2007. doi: 10.1109/ITWITWN.2007.4318051.  Google Scholar [6] E. Thomas and F. Oggier, A note on quasi-uniform distributions and abelian group representability, in 2012 International Conference on Signal Processing and Communications, Bangalore, India, 2012. doi: 10.1109/SPCOM.2012.6290020.  Google Scholar [7] X. Yan, R. Yeung and Z. Zhang, The capacity for multi-source multi-sink network coding, in 2007 International Symposium on Information Theory, Nice, France, 2007. doi: 10.1109/ISIT.2007.4557213.  Google Scholar
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