In this paper, in addition to some elementary facts about the ultra-GROUPS, which their structure based on the properties of the transversal of a subgroup of a group, we focus on the relation between a group and an ultra-group. It is verified that every group is an ultra-group, but the converse is not true generally. We present the conditions under which, for every normal subultra-group of an ultra-group over a group, there exists a normal subgroup of that certain group. Moreover, by proving this feature that a monomorphism in GROUPS preserve the ultra-GROUPS over GROUPS, we show that, corresponding to any monomorphism in GROUPS, there is an ultragroup homomorphism on the subGROUPS of those GROUPS. Finally, we prove isomorphism theorems for ultra-GROUPS. These theorems connect three notions subultra-group, normal subultra-group, quotient ultra-group, and directly, similar to the isomorphism theorems in group theory and module theory are proved.

In their paper "Finite GROUPS whose noncentral commuting elements have centralizers of equal size", S. Dolfi, M. Herzog and E. Jabara classify the GROUPS in question- which they call CH-GROUPS- up to finite p-GROUPS. Our goal is to investigate the finite p-GROUPS in the class. The chief result is that a finite p-group that is a CH -group either has an abelian maximal subgroup or is of class at most p+1. Detailed descriptions, in some cases characterisations up to isoclinism, are given.

This article discusses the focus GROUPS as a useful method in qualitative research. Focus GROUPS are less discussed in research sources, while they have many applications in qualitative researches. In this article, the origin, history, definition, and application of the focus GROUPS are studied. Also, the method of participant selection, the size and count of GROUPS, method of questioning, survey size, survey type, moderator specifications, collecting and analyzing data in focus GROUPS are discussed.

For a  nite group G and a positive integer n, let G(n) be the set of all elements in G such that xn = 1. The GROUPS G and H are said to be of the same (order) type if jG(n)j = jH(n)j, for all n. The main aim of this paper is to show that if G is a  nite group of the same type as Suzuki GROUPS Sz(q), where q = 22m+1  8, then G is isomorphic to Sz(q). This addresses to the well-known J. G. Thompson's problem (1987) for simple GROUPS.

We record for reference a detailed description of the automorphism GROUPS of the GROUPS of order p 2 q, where p and q are distinct primes.