A many identities group (MI-group, for short) is an algebraic structure which is generalized a monoid with cancellation laws and is endowed with an invertible anti-automorphism representing inversion. In other words, an MI-group is an algebraic structure generalizing the group concept, except most of the elements have no inverse element. The con-cept of a topological MI-group, as a preliminary study, in the paper " Topological MI-group: Initial study" was introduced by M. Holacapek and N. Skorupova, and we have given a more comprehensive study of this concept in our two recent papers. This article is a continuation of the effort to develop the theory of topological MI-groups and is focused on the study of separation axioms and the isomorphism theorems for topological MI-groups. Moreover, some conditions under which a MI-subgroup is closed will be investigated, and finally, the existence of nonnegative invariant measures on the locally compact MI-groups are introduced.

A group G is called metahamiltonian if all its non-abelian subgroups are normal. The aim of this paper is to provide an updated survey of research concerning certain classes of generalized metahamiltonian groups, in various contexts, and to prove some new results. Some open problems are listed.

Let G be a finite group and let NG denote the set of all non- trivial proper subgroups of G, A member K of NG is called n-decomposable if K is the union of n distinct conjugacy classes of G. The group G is called n-decomposable if NG≠ǿ and every member of NG is n decomposable. In this paper we investigate the structure of n-decomposable finite groups for n ≤ 6 and classify them in the class of finite non-perfect groups.

‎In this paper‎, ‎we construct a hypergroup by using a hypergroup‎ ‎$(H,\circ)$ and a polygroup $(P,\cdot)$‎, ‎and call it‎ ‎$(H,Poly(P))$-hypergroup‎. ‎The method of constructing hypergroups in this paper is not present in the established techniques of group theory‎. ‎Moreover‎, ‎we compare‎ ‎$(H,Poly(P))$-hypergroups with $K_H$-hypergroups‎, ‎complete‎ ‎hypergroups and extensions of polygroups by polygroups‎.