On Certain F-Perfect Groups

We consider certain groups which have no proper subgroup of finite index such as minimal non-soluble p-groups and give some descriptions of them.

A group which has no proper subgroup of finite index is called -perfect. Let  be a class of groups.  If a group G is not in  but every proper subgroup of G is in   then G is called a minimal non -group and denoted usually by . Clearly every perfect p-group for some prime p is -perfect and the structure of many -groups is investigated in perfect p-case.

In the present article we mostly take , the class of all soluble groups, as the class , i.e. we consider certain  -groups. It not known yet that if locally finite -p-group which is not finitely generated exist. Such groups are mainly studied in [5], [4], [2], [3] (in some general form) and given certain descriptions.

Let G be a group and H be a subgroup of G. If  is infinite, 1 and for every proper subgroup K of G, Ç is finite, then G is called a barely transitive group and H is called a point stabilizer. Though thr definition of barely transitive groups has permutation groups origin, we use above abstract definition of these groups (see [7]).

We give certain applications of Khukhro-Makarenko Theorem to -groups. Also we provide a corollary to [8,Satz 6] which is used effectively in most of the cited  articles and give certain applications of it. Finally we define Weak Fitting groups and  a useful class  of groups and give certain results related to these notions.

Key Words:-group,  -groups, -Perfect groups, barely transitive group.