In this paper, we review two properties of completeness known as the Bourbaki-completeness and cofinal Bourbaki-completeness in the setting of metric spaces. These notions came from new classes of generalized Cauchy sequences appearing when characterizing the so-called Bourbaki-boundedness in a similar way that Cauchy sequences characterize the totally boundedness. For the clustering of Bourbaki–Cauchy sequences and cofinally Bourbaki–Cauchy sequences, we have respectively what is call Bourbaki-completeness and cofinal Bourbaki-completeness of metric spaces. The topological problem of metrizability by means of a Bourbaki-complete or a cofinally Bourbaki-complete metric has also been considered. Finally, we present detailed review of some relationships and mutual differences between these kinds of completeness.

Frattini subgroup, Φ(G), of a group G is the intersection of all the maximal subgroups of G, or else G itself if G has no maximal subgroups. If G is a p-group, then Φ(G) is the smallest normal subgroup N such the quotient group G/N is an elementary abelian group. It is against this background that the concept of p-subgroup and fitting subgroup play a significant role in determining Frattini subgroup (especially its order) of dihedral groups. A lot of scholars have written on Frattini subgroup, but no substantial relationship has so far been identified between the parent group G and its Frattini subgroup Φ(G) which this tries to establish using the approach of Jelten B. Napthali who determined some internal properties of non abelian groups where the centre Z(G) takes its maximum size.