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.

p'] satisfies all the hypotheses of our corollary for the normal subgroup H/[H.

On the other hand, given a group G with a

normal subgroup N [?

For example, Skiba proved: Let E be a normal subgroup of a finite group G.

Let p be a prime dividing the order of a group G with ([absolute value of G], p - 1) = 1 and H a normal subgroup of G such that G/H is p-nilpotent.

If(K, *) is an idempotent quasigroup and the right nucleus of(Q, *) is a

normal subgroup, then the orbits of the right-regular permutation group are normal subquasigroups ofthe kernel quasigroups ofthe extension.

If there exists a

normal subgroup N in G such that G/N is p--nilpotent and each subgroups of N of order [p.

Since in any topological group, the intersection of all open neighborhoods of [member of] is a

normal subgroup, thus V(e) is a

normal subgroup of [member of] such that V(g) = gV(e) (g [member of] G), moreover for each g [member of] V(e) we have V(g) = V(e).

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.