American Institute of Mathematical Sciences

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

On Abelian group representability of finite groups

Eldho K. Thomas 1, , Nadya Markin 1, and  Frédérique Oggier 1,

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.
Keywords: group representability, Nilpotent group, entropic vector..
Mathematics Subject Classification: Primary: 20D15; Secondary: 94A1.
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

show all references

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

[1]

Mickaël D. Chekroun, Jean Roux. Homeomorphisms group of normed vector space: Conjugacy problems and the Koopman operator. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 3957-3980. doi: 10.3934/dcds.2013.33.3957
[2]

Sergio Estrada, J. R. García-Rozas, Justo Peralta, E. Sánchez-García. Group convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 83-94. doi: 10.3934/amc.2008.2.83
[3]

Heping Liu, Yu Liu. Refinable functions on the Heisenberg group. Communications on Pure & Applied Analysis, 2007, 6 (3) : 775-787. doi: 10.3934/cpaa.2007.6.775
[4]

Stefan Haller, Tomasz Rybicki, Josef Teichmann. Smooth perfectness for the group of diffeomorphisms. Journal of Geometric Mechanics, 2013, 5 (3) : 281-294. doi: 10.3934/jgm.2013.5.281
[5]

Van Cyr, John Franks, Bryna Kra, Samuel Petite. Distortion and the automorphism group of a shift. Journal of Modern Dynamics, 2018, 13: 147-161. doi: 10.3934/jmd.2018015
[6]

Woochul Jung, Keonhee Lee, Carlos Morales, Jumi Oh. Rigidity of random group actions. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6845-6854. doi: 10.3934/dcds.2020130
[7]

Daniele D'angeli, Alfredo Donno, Michel Matter, Tatiana Nagnibeda. Schreier graphs of the Basilica group. Journal of Modern Dynamics, 2010, 4 (1) : 167-205. doi: 10.3934/jmd.2010.4.167
[8]

Kesong Yan, Qian Liu, Fanping Zeng. Classification of transitive group actions. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5579-5607. doi: 10.3934/dcds.2021089
[9]

Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113
[10]

Yves Guivarc'h. On the spectrum of a large subgroup of a semisimple group. Journal of Modern Dynamics, 2008, 2 (1) : 15-42. doi: 10.3934/jmd.2008.2.15
[11]

Marcelo Sobottka. Topological quasi-group shifts. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 77-93. doi: 10.3934/dcds.2007.17.77
[12]

Dandan Cheng, Qian Hao, Zhiming Li. Scale pressure for amenable group actions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1091-1102. doi: 10.3934/cpaa.2021008
[13]

Jean-Francois Bertazzon. Symbolic approach and induction in the Heisenberg group. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1209-1229. doi: 10.3934/dcds.2012.32.1209
[14]

Nadya Markin, Eldho K. Thomas, Frédérique Oggier. On group violations of inequalities in five subgroups. Advances in Mathematics of Communications, 2016, 10 (4) : 871-893. doi: 10.3934/amc.2016047
[15]

Katarzyna Grabowska, Marcin Zając. The Tulczyjew triple in mechanics on a Lie group. Journal of Geometric Mechanics, 2016, 8 (4) : 413-435. doi: 10.3934/jgm.2016014
[16]

Maik Gröger, Olga Lukina. Measures and stabilizers of group Cantor actions. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2001-2029. doi: 10.3934/dcds.2020350
[17]

Tao Yu, Guohua Zhang, Ruifeng Zhang. Discrete spectrum for amenable group actions. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5871-5886. doi: 10.3934/dcds.2021099
[18]

Yuri Berest, Alimjon Eshmatov, Farkhod Eshmatov. On subgroups of the Dixmier group and Calogero-Moser spaces. Electronic Research Announcements, 2011, 18: 12-21. doi: 10.3934/era.2011.18.12
[19]

Mike Boyle, Sompong Chuysurichay. The mapping class group of a shift of finite type. Journal of Modern Dynamics, 2018, 13: 115-145. doi: 10.3934/jmd.2018014
[20]

Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormalization group transformation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 437-446. doi: 10.3934/dcdsb.2016.21.437

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (134)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy