The Lattice Structure Of The Subgroups Of Order 8 In The Subgroup Lattices Of 3 X 3 Matrices Over Z2

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

doi:https://doi.org/10.14445/22315373/IJMTT-V66I8P508

International Journal of Mathematics Trends and Technology

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Volume 66 | Issue 8 | Year 2020 | Article Id. IJMTT-V66I8P508 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I8P508

The Lattice Structure Of The Subgroups Of Order 8 In The Subgroup Lattices Of 3 X 3 Matrices Over Z2

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

Abstract

Let G be the set of all 3 * 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-p)(pn-p)(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-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 8 of L(G) in the case when P=2.

Keywords

Matrix group, Subgroups, Poset, Lattice, Atom.

References

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[3] Gardiner. C.F, A first course in group theory, Springer-Verlag, Berlin,1997.
[4] Gratzer. G, General Lattice theory : Birkhauser Veslag, Basel,1998.
[5] Herstien I.N, Topics in Algebra, John Wiley and sons, New York,1975.

Citation :

V.Durai Murugan, R. Seethalakshmi, Dr.P.Namasivayam, "The Lattice Structure Of The Subgroups Of Order 8 In The Subgroup Lattices Of 3 X 3 Matrices Over Z2," International Journal of Mathematics Trends and Technology (IJMTT), vol. 66, no. 8, pp. 72-81, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I8P508