In this paper we highlight the connection between certain classes of REGULAR SUBGROUPs of the affine group AGLn (F), F a field, and associative nilpotent F-algebras of dimension n. We also describe how the classification of projective congruence classes of square matrices is equivalent to the classification of REGULAR SUBGROUPs of particular shape.

We introduce a new REGULAR relation δ on a given group G and show that δ is a congurence relation on G, with respect to module the commutator SUBGROUP of G. Then we show that the effect of this relation on the fundamental relation β is equal to the fundamental relation γ, and we conclude that, if ρ is an arbitrary strongly REGULAR relation on the hypergroup H, then the effect of δ on ρ, results in a strongly REGULAR relation on H such that its quotient is an abelian group.

A result of Dixon, Evans and Smith shows that if G is a locally (soluble-by- finite) group whose proper SUBGROUPs are (finite rank) -by-abelian, then G itself has this property, i.e. the commutator SUBGROUP of G has finite rank. It is proved here that if G is a locally (soluble-by- finite) group whose proper SUBGROUPs have minimax commutator SUBGROUP, then also the commutator SUBGROUP G’ of G is minimax. A corresponding result is proved for groups in which the commutator SUBGROUP of every proper SUBGROUP has finite torsion-free rank.

Hypergroups that have at least one identity element and where each element has at least one inverse are called REGULAR hypergroup. In this regards, for a REGULAR hypergroup $H$, it is shown that there exists a correspondence between the set of all strongly REGULAR relations on $H$ and the set of all normal subhypergroups of $H$ containing $S_{\beta}$. More precisely, it has been proven that for every strongly REGULAR relation $\rho$ on $H$, there exists a unique normal subhypergroup of $H$ containing $S_{\beta}$, such that its quotient is a group, isomorphic to $H/\rho$. Furthermore, this correspondence is extended to a lattice isomorphism between them.

Let $G$ be a group and $S$ be the collection of all non-trivial proper SUBGROUPs of $G$. The co-maximal SUBGROUP graph $\Gamma(G)$ of a group $G$ is defined to be a graph with $S$ as the set of vertices and two distinct vertices $H$ and $K$ are adjacent if and only if $HK=G$. In this paper, we study the comaximal SUBGROUP graph on finite dihedral groups. In particular, we study order, maximum and minimum degree, diameter, girth, domination number, chromatic number and perfectness of comaximal SUBGROUP graph of dihedral groups. Moreover, we prove some isomorphism results on comaximal SUBGROUP graph of dihedral groups.

We prove that the class of pro finite groups G that have a factorization G=AB with A and Babelian closed SUBGROUPs, is closed under taking inverse limits of surjective inverse systems. This is a generalization of a recent result by K.H. Hofmann and F.G. Russo. As an application we reprove their generalization of Iwasawa’s structure theorem for quasihamiltonian pro-p groups.

In this paper, we study the notion of solvable L-SUBGROUP of an L-group and provide its level subset characterization and this justifies the suitability of this extension. Throughout this work, we have used normality of an L-SUBGROUP of an L-group in the sense of Wu rather than Liu.