We prove that the following families of (in, nite) groups have Complemented subgroup lattice: alternating groups, , nitary symmetric groups, Suzuki groups over an in, nite locally , nite , eld of characteristic 2, Ree groups over an in, nite locally , nite , eld of characteristic 3. We also show that if the Sylow primary subgroups of a locally , nite simple group G have Complemented subgroup lattice, then this is also the case for G.

P. Hall proved that a , nite group G is supersoluble with elementary abelian Sylow subgroups if and only if every subgroup of G is Complemented in G. He called such groups Complemented. A. Ballester-Bolinches and X. Guo established the structure of minimal non-Complemented groups. We give the classi, cation of , nite non-soluble groups all of whose second maximal subgroups are Complemented groups. We also prove that every , nite group with less than 21 non-Complemented non-minimal f 2,3,5 g-subgroups is soluble.

The notion of hemiComplemented lattices is introduced and some of the properties of these algebras are studied. Some characterization theorems of hemiComplemented lattices are derived with the help of minimal prime D-filters, ideals, and congruences. The notion of D-Stone lattices is introduced and then derived a set of equivalent conditions for a hemiComplemented lattice to become a D-Stone lattice. Hemicom-Complemented lattices and D-Stone lattices are characterized in topological terms.

The notion of hemiComplemented lattices is introduced and some of the properties of these algebras are studied. Some characterization theorems of hemiComplemented lattices are derived with the help of minimal prime D-filters, ideals, and congruences. The notion of D-Stone lattices is introduced and then derived a set of equivalent conditions for a hemiComplemented lattice to become a D-Stone lattice. Hemicom-Complemented lattices and D-Stone lattices are characterized in topological terms.

In this paper, we introduced the concepts of normlet and normal ideal in a pseudo-Complemented almost distributive lattice and studied its properties. We have char-acterized normal ideals and established equivalent conditions for every ideal to become a normal ideal. Also, derived equivalent conditions for the set of all prime normal ideals of a pseudo-Complemented ADL to become a Hausdorff space.

Let L be a  nite-dimensional Lie algebra. We say a subalgebra H of L is per-mutably Complemented in L if there is a subalgebra K of L such that L = H + K and H \K = 0. Also, if every subalgebra of L is permutably Complemented in L, then L is called completely factorisable. In this article, we consider the in uence of these concepts on the structure of a Lie algebra, in particular, we obtain some characterizations for supersolvability of a  nite-dimensional Lie algebra in terms of permutably Complemented subalgebras.

Let G be a finite group. A subgroup H of G is called an H-subgroup in G if NG (H) Ç Hg £ H for all g  Î G.A subgroup H of G is called a weakly H*-subgroup in G if there exists a subgroup K of G such that G=HK and H Ç K is an H-subgroup in G. We investigate the structure of the finite group G under the assumption that every cyclic subgroup of G of prime order p or of order 4 (if p=2) is a weakly H*-subgroup in G. Our results improve and extend a series of recent results in the literature.

A , nite group G, in which two randomly chosen subgroups H and K commute, has been classi, ed by Iwasawa in 1941. It is possible to de, ne a probabilistic notion, which \measures the distance" of G from the groups of Iwasawa. Here we introduce the generalized subgroup commutativity degree gsd(G) for two arbitrary sublattices S(G) and T(G) of the lattice of subgroups L(G) of G. Upper and lower bounds for gsd(G) are shown and we study the behaviour of gsd(G) with respect to subgroups and quotients, showing new numerical restrictions.

In this paper, we give a complete proof of Theorem 4.1 (ii) and a new elementary proof of Theorem 4.1 (i) in [Li and Shen, On the intersection of the normalizers of the derived subgroups of all subgroups of a nite group, J. Algebra, 323 (2010) 1349-1357]. In addition, we also give a generalization of Baer's Theorem.