The Lattice Structure of The Subgroups of
Order 27 In The Subgroup Lattices of 3X3
Matrices Over Z3

V. Durai murugan; R. Seethalakshmi; Dr.P.Namasivayam

doi:https://doi.org/10.14445/22315373/IJMTT-V66I9P505

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 66 | Issue 9 | Year 2020 | Article Id. IJMTT-V66I9P505 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I9P505

The Lattice Structure of The Subgroups of Order 27 In The Subgroup Lattices of 3X3 Matrices Over Z3

V. Durai murugan, R. Seethalakshmi, Dr.P.Namasivayam

Abstract

Let G be the set of all 3 X 3 non-singular matrices (a b c d e f g h i), where a,b,c,d,e,f,g,h,i are integers modulo p. Then G is a group under matrix multiplication modulo p, of order ( pn-1)(pn-p2).. ...(pn-pn-1). Let G be the subgroup of G defined by G = { (a b c d e f g h i) ε G : | a b c d e f g h i| = 1}. Then G is of order (pn-1)(pn-p)(pn-p2)....(pn-pn-1)/p-1. Let L(G) be the lattice formed by all subgroups G. In this paper, we give the structure of the subgroups of order 27 of L(G) in the case when P=3.

Keywords

Matrix group, subgroups, Lagrange’s theorem, Lattice, Atom.

References

[1] Aubert : Necessary and sufficient conditions for isotropic rank one functions in dimension 2; Journal of Elasticity, 39, 31-46 (1995).
[2] B. Dacorogna., "Necessary and sufficient conditions for strong ellipticity of isotropic functions in any dimension". Discrete and continuous dynamical systems-series B, Vol 1, Number 2, pp. 257-263 (2001).
[3] E. Diouf. " Sur l’ellipticité des équations non linéaires pour une certaine classe de matériau isotrope et compressible". International Journal of Innovation and Applied Studies, ISSN 2028-9324 Vol. 13 No. 2 Oct., pp. 327-334 (2015).
[4] A. Lyapunov. "The general problem of the stability of motion". CRC Press, (ISBN 0748400621), (1992).
[5] J. M. Ball. ?Convexity conditions and existence theorems in nonlinear elasticity". Arch. Rat. Mech. Anal. 63, pp. 337?403 (1977).
[6] I. D. Ghiba, R. G. martin, P. Neff. "Rank-one convexity implies polyconvexity in isotropic planar incompressible elasticity". arXiv:1609.01339v1 [math.CA], 5 Sep 2016.
[7] Hjørungnes. "Complex-Valued Matrix Derivatives: With Applications in Signal Processing and Communications". CUP, p. 121- , (2011). 9

Citation :

V. Durai murugan, R. Seethalakshmi, Dr.P.Namasivayam, "The Lattice Structure of The Subgroups of Order 27 In The Subgroup Lattices of 3X3 Matrices Over Z3," International Journal of Mathematics Trends and Technology (IJMTT), vol. 66, no. 9, pp. 26-36, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I9P505