Life Science Journal 2012;9(4) http://www.lifesciencesite.com 2387 Finite Groups With At Most Nine Non T-Subgroups Muhammad Arif 1 , Muhammad Shah 1 , Saeed Islam 1 , Khalid Khan 2 1 Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan 2 Department of Science and Information Technology Govt; Peshawar, Pakistan marifmaths@awkum.edu.pk (M. Arif), shahmaths_problem@hotmail.com (M. Shah), proud_pak@hotmail.com (S. Islam), khalidsa02@gmail.com (K. Khan) Abstract. In this short note, we extend the characterization of soluble groups by using the number of their non-T-subgroups and also we classify finite groups having exactly nine non-T-subgroups. [Arif M, Shah M, Islam S, Khan K. Finite Groups with at most nine non T-Subgroups. Life Sci J 2012;9(4):2387-2389] (ISSN:1097-8135). http://www.lifesciencesite.com . 354 Key words. Finite soluble groups, Normal subgroups, T-subgroups 1. Introduction Group is an algebraic structure provides information about operations and relations satisfying certain algebraic conditions. Group and its algebraic properties provides many application in various fields like chemisty [8], Physics [9] and Computer sciences [10-23]. In this paper we concentrate on A group G is T- group when its every subnormal subgroup is normal. Thus T-groups are exactly those groups in which normality is a transitive relation. This structure has been studied by many authors such as Gaschutz [5], Zacher [4] and Robinson [6]. Robinson characterized finite soluble groups by T-groups. He proved that if all subgroups of a finite group are T-groups then G is soluble. For details we refer to Robinson [2]. Arif et all characterized finite soluble groups by number of non-T-subgroups. In [6] this has been proven that if a finite group G has at most 4 non-T-subgroups then the group will be soluble. It has also been shown there that finite groups having exactly five non-T-subgroups may not be soluble. The purpose of this present note is to extend the characterization of finite groups up to 9 non-T-subgroups of group G . Thus we prove if G is a finite group all whose proper subgroups are T-groups except for 6, 7, 8, n  then G is soluble see Theorem 1. We also prove that if G is a finite group such that G contains exactly 9 non-T-subgroups then either G is soluble or G is isomorphic to (2,8) SL see Theorem 2. Since our characterization is based on the number of non- T-subgroups so in Table 1 we present the details of the number of non-T-subgroups of each non-soluble group up to order 900 for the sake of completeness and for future use. We have used the following two GAP [3] functions to construct this table. In Table 1, O denotes the order of non-soluble groups, 1 N denotes the number of non-soluble groups of the order given in column first and 2 N denotes the number of non-T-subgroups of each non-soluble subgroup of the group given in column second. function( G ):= NrNonTSubgroups local allsubgrps, i; allsubgrps:=Subgroups( G ); for i in allsubgrps do if IsSubnormal(G,i) then if not IsNormal(G,i) then return false; break; fi; fi; od; return true; end ; function( G ):= NrNonTSubgroups local allsubgrps; allsubgrps:= Subgroups( G ); return Number(allsubgrps, t  not IsTGroup( t )); end; 1. Preliminaries The following results from [7] are needed. (1). If G is a finite simple group such that G has exactly n non-T-subgroups ( 1) n  , then G is isomorphic to a subgroup of n S [7, Theorem 2.2]. (2). Let G be a finite group with exactly n non-T-subgroups and suppose that any finite group with exactly m non-T-subgroups is soluble for 1 1 m n    [7, Theorem 2.3].