Given a prime p, a group is called residually p if the intersection of its p-power index

normal subgroups is trivial, and called virtually residually p if it has a finite index subgroup that is residually p.

Thomas: A complete study of the lattices of fuzzy congruences and

normal subgroups, Information Sciences, 82(1995), 197-218.

Topics covered include counting of subgroups and proof of the main counting theorems, regular p-groups and regularity criteria, p-groups of maximal class and their characterizations, characters of p-groups, p-groups with large Schur multiplier and commutator subgroups, (p--1)- admissible Hall chains in

normal subgroups, powerful p-groups, automorphisms of p-groups, p-groups that have nonnormal subgroups that are all cyclic, and Alberin's problem of abelian subgroups of small index.

Papers cover such subjects as outer automorphism groups of certain orientable Seifert three-manifold groups, a proposed public key cryptosystem using the modular group,

normal subgroups of themodular group and other Hecke groups, unions of varieties and quasi-varieties, context-free irreducible word problems in groups, informative words and discreteness, using group theory for knowledge representation and discovery, torsion in maximal arithmetic Fuchsian groups, density of test elements in finite Abelian groups and the Rosenberg "monster.

It is well known that there exists a one to one correspondence between Alexandroff topologies on group [member of] which made [member of] a topological group and its normal subgroups ([4], theorem 4), moreover if for each normal subgroup N of G, [T.

There exists a one to one correspondence between functional Alexandroff topologies on group [member of] which made [member of] a topological group and its finite normal subgroups.

G, then G/H [member of] F, and (2) if G/M and G/N are in F, then G/M [intersection] N is in F for normal subgroups M and N of G.

2) P/[PHI](P)is a minimal normal subgroup of G/[PHI](P).

ii) P/[PHI](P) is a minimal normal subgroup of G/[PHI](P).

Let p be the smallest prime dividing the order of a group G and H a normal subgroup of G such that G/H is p-nilpotent.

Given a group G suppose it has a

normal subgroup N [?

For example, Ezquerrohas proved: Let G be a group with a

normal subgroup H such that G/H is supersolvable.