Classes of groups defined by series or normal factor coverings

Abstract

Of primary interest in this work is the establishing of new closure properties for certain classes of groups defined by means of series or normal factor coverings. (Here, series is as defined by Robinson [IO, p. 91, and norma factor covering is as defined by Durbin [3].) We begin by identifying cxactl!; those classes of groups with which we shall deal. After the identification of these classes, we discuss the new closure properties. Let r’. be a variety of groups defined by a set of words CIT. Also, let pW be the subgroup theoretical property IV-margina [IO, p. 91. Then the series classes with which we deal are @V, P,V and (PI&‘)‘), , the latter being defined by G E ( pW)s if and only if G has ap W-series. (The operations P, fi, and the notion of a pI&‘-series are defined by Robinson [ 10, pp. 12-131.) These classes receive attention in [I I] and are characterized in an elegant fashion by Hickin and Phillips in a recent work [5]. 1% :e note that some familiar classes result whenever W consists of the commutator word. For then ‘Y is the v-ariet!of all abelian groups, and the classes of SN-groups, S-groups, Z-groups (in the sense of Kurosh [6]) are given by i%-, @,,-l ‘, ( pW)s , respectively. Now let us define some classes of groups by means of normal factor coverings. Define an operation F, by G t F,X if and only if G has a normal factor covering with X-factors, where 3 is any class of groups. If I. is a variety of groups defined bv a set of words IX, then in the sequel we shall be concerned with the normai factor covering classes F%Y“ and (pW), , which is defined by G E ( pW), provided G has a normal factor covering {rC, , H, : a: E A] such that H%/K,, is W-marginal in G/K, , for each a E A. Whenever W consists of the commutator word, we mention that the classes F,‘Y . and (pW), have appeared in the literature [I, 2, 31.