Configuration of Groups Containing Generalized Normal Subgroups

This chapter provides a short mathematical introduction to group and subgroup algebra. It is constructed in the following order: definitions, propositions, and proofs. The concepts and terminology provided here will serve as the foundation for the chapters that follow, which will deal with group theory in its tighter sense and its application to physics problems. Mathematics prerequisites are at the bachelor's level. 1 A subgroup X of a group G is said to be virtually normal if it contains a finite number of conjugates, or equivalently, if the normaliser  NG(X) of X has a finite index in G. B.H. Neumann famously proved that all subgroups of a group G are practically normal if and only if the centre Z(G) has a finite index in G. I.I Eremin [4] expanded on this theory, demonstrating that a subgroup X of a group G is nearly normal if it has a finite index in its normal closure, . In the aforementioned publication, Neumann demonstrated that all subgroups of a group G are nearly normal if and only if the commutator subgroup G' of G is finite. It is sufficient to assume that all abelian subgroups are nearly normal (a result that was obtained by M.J.Tomkinson [5]). In general, the concepts of almost normal and nearly normal subgroups are incomparable. For instance, all subgroups of the base group of the standard wreath product W= are almost normal in W, but only a few of them are nearly normal. Similarly, if G is any group containing an infinite minimal normal subgroup N, which is abelian of prime exponent, then each proper subgroup of finite index of N is nearly normal in G but has infinitely many conjugates.On the other hand, it follows from Neumann's work and the renowned theorem of Schur on the finiteness of the commutator subgroup of central-by-finite groups (see, for example,[7] Part 1, Theorem 4.12) that if all subgroups of a group G are almost normal, then they must be nearly normal.It should also be noted that the structures of groups in which every infinite subgroup is almost normal and groups in which all infinite subgroups are nearly normal have been discussed separately. It turns out that knowing that all subgroups of a certain type are virtually normal (or that they are all almost normal) is useful.The major purpose of this paper is to analyse groups in which every member of a relevant system of subgroups is either almost normal or nearly normal; in particular, Tomkinson's theorem mentioned above will be generalised to this case.

Y.D. Polovicki result demonstrates that a group G has finitely many normalisers of abelian subgroups if and only if it is finite in the centre. In recent years, many more studies have appeared on the structure of groups with finitely many normalisers of subgroups with a certain property (see for example [2],[8],[9],[10],[19]); We will also look at groups with a finite number of normalisers for subgroups that are neither virtually normal nor nearly normal.

Remember that the FC-center of a group G is the subgroup that contains all elements with finitely many conjugates, and G is a F C -group if it coincides with its FC-Centre. The theory of FC-groups is relevant in many concerns about infinite groups with finiteness criteria; the monograph contains the main features of FC-groups.

It is easy  to show that a cyclic subgroup  of a group G is almost normal if and only if it is nearly normal, and both such features are also equivalent to the fact that the conjugacy class of x in G is finite. Thus a group G is an FC-group if and only if all its cyclic subgroups are almost normal(or nearly normal). It turns out that in the universe of FC-groups, approximately normalcy is a stronger feature than nearly normality.

Lemma: Let G be an FC-group, and let X be a nearly normal subgroup of G. Then X is approximately normal in G.

Proof: As the normalize  NG(X) has finite index in G ,there exits a finitely generated subgroup E of G such that G =hE, NG(X) i. Moreover ,the subgroup E can be chosen normal in G, since its normal closure EG is like –wise finitely generated .Thus G = NG(X) E and so XG =XE. Clearly ,E is central –by-finite ,so that it satisfies the maximal condition on subgroups and in particular its subgroup[X,E] is finitely generated .On the other hand , the commutator subgroup  of G is locally finite and hence [X,E] is finite .Therefore the subgroup X has finite index in its normal closure XG =X[X,E] and so it is nearly normal in G.

The main result of this section is an extension of Tomkinson's theorem from the introduction to the case in which every (abelian) subgroup is either almost normal or nearly normal; it will be obtained as a result of a theorem on groups with finitely many normalisers of subgroups with a suitable property. For this reason, we require a result of B.H. Nesumann, which holds in the more general case of groups covered by cosets of subgroups.

The main result of this section is an extension of Tomkinson's theorem quoted in the introduction to the case in which every (abelian) subgroup is either almost normal or nearly normal; it is obtained as a result of a theorem on groups with finitely many normalisers of subgroups with a suitable property. For this reason, we need B.H.Nesumann's conclusion, which holds in the more general case of groups covered by subgroup cosets.

Section 4  stated that any virtually normal subgroup of an FC-Group is nearly normal. Our next lemma demonstrates that the contrary result holds in the case of groups with finite Prufer rank.

Thus, a subgroup X of a group G is almost normal if its index /G: NG(X) is finite, but X is nearly normal if it has a finite index in the normal closure XG. This work looks at the structure of groups in which each (infinite) subgroup is either almost normal or nearly normal.