Subgroup Graphs of Finite Groups
(1) Usmanu Danfodiyo University, Sokoto, Nigeria.
(2) Usmanu Danfodiyo University, Sokoto, Nigeria.
(3) Usmanu Danfodiyo University, Sokoto, Nigeria.
(*) Corresponding Author
Abstract
Let G be a fnite group with the set of subgroups of G denoted by S(G), then the subgroup graphs of G denoted by T(G) is a graph which set of vertices is S(G) such that two vertices H, K in S(G) (H not equal to K)
are adjacent if either H is a subgroup of K or K is a subgroup of H. In this paper, we introduce the Subgroup
graphs T associated with G. We investigate some algebraic properties and combinatorial structures of Subgroup
graph T(G) and obtain that the subgroup graph T(G) of G is never bipartite. Further, we show isomorphism
and homomorphism of the Subgroup graphs of finite groups.
Let be a finite group with the set of subgroups of denoted by , then the subgroup graphs of denoted by is a graph which set of vertices is such that two vertices , are adjacent if either is a subgroup of or is a subgroup of . In this paper, we introduce the Subgroup graphs associated with . We investigate some algebraic properties and combinatorial structures of Subgroup graph and obtain that the subgroup graph of is never bipartite. Further, we show isomorphism and homomorphism of the Subgroup graphs of finite groups.
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I. Kleiner, History of Group theory, History of Abstract Algebra, Birkhauser Boston, 17-39, (2007).
J. Zhang, F. Xiong and J. Kang, The application of Group theory in communication operation pipeline system, Mathematical problems in Engineering, (2018).
J. Laane and E. J. Ocola, Application of symmetry and group theory for the investigation of molecular vibrations, Acta Applicandae Mathematicae, 118(1), 3 24, (2012).
E. A. Rietman, R. L. Karp and J. A. Tuszynski, Review and application of group theory to molecular system biology, Theoretical Biology and medical modeling, 8(21), (2011).
H. Osborn, Symmetry relationships between Crystal Structures: application of crystallographic group theory in crystal chem- istry, Contemporary Physics, 6(1), 97-98, (2015).
A. Cayley, Desiderata and suggestions: The theory of groups: graphical representation, American Journal of Mathematics, 1 (2) (1878) 403-405.
W. B. Vasantha Kandasamy and F. Samarandache, Groups as Graphs, Editura cuart and authors, (2009).
P. H. Zieschang, Cayley graph of finite groups, Journal of Algebra, 118, 447 - 454 (1988).
P. J. Cameron and S. Ghosh, The power graph of finite groups, Discrete Mathematics, 311, 1220 -1222, (2011).
S. U. Rehman, A. Q. Baig, M. Imran and Z. U. Khan, Order divisor graphs of finite groups, An. St. Ovidus Constanta, 26(3), 29-40, (2018).
J. S. Williams, Prime graph components of finite groups, Journal of Algebra, 69(2), 487-513, (1981).
X. L. Ma, H. Q. Wei and G. Zhong, The cyclic graph of a finite groups, Algebra, (2013).
A. Erfanian and B. Tolue, Conjugate graphs of finite groups, Discrete Mathematics, Algorithm and Applications, 4(2), (2012).
B. Akbari, Hall graph of a finite group, Note Mat., 39(2), 25-37, (2019).
A. Lucchini, The independence graph of a finite group, Monatsheft Fur Mathematik, 193, 845-856, (2020).
D. Gorenstein, Finite Groups, Harper & Row, New York, 1968.
D. J. S. Robinson, A course in the theory of Groups, 2nd edition, Springer-Verlag, New York, (1996).
S. D. David and M. F. Richard, Abstract algebra, 3rd Edition, John Wiley and Son Inc., (2004).
C. Godsil and G. Boyle, Algebraic graph theory, 5th edition, Springer, Boston New York, (2001).
A. Gupta, Discrete mathematics, S.K. Kataria & Sons, 258-310, 2008.
P. J. Cameron, Notes on finite group theory, www.maths.qmul.ac.uk (2013).
H. E. Rose, A Course on Finite Groups; Springer Science & Business Media,(2009).
A. E. Clement, S. Majewicz and M. Zyman, Introduction to Nilpotent Groups; The Theory of Nilpotent Group. Birkhauser, Cham, (2017).
W. A. Trybulec, Commutator and Center of a Group, Formalized Mathematics, Universite Catholique de Louvain, 2(4)(1991).
R. M. Guralnick, Commutators and Commutator Subgroups, Advances in Mathematics, 45, 319-330 (1982).
people.math.binghamton.edu (Accessed on 2nd October, 2020).
www.math.columbia.edu (Accessed 6th November, 2020).
K. Conrad Simplicity of An; kconrad.math.uconn.edu (Accessed on 7th November, 2020).
S. R. Cavior, The Subgroups of the Dihedral groups; Mathematics Magazine, 48, 107, (1975).
M. Tarnauceanu, A characterization of the quaternion group, An. St. Univ. Ovidius Constanta, 21(1), 209-2014, (2013).
DOI: https://doi.org/10.24071/ijasst.v3i2.3765
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