Finite groups with systems of I£-embedded subgroups

In recent years, a series of papers about cover-avoiding property of subgroups appeared and all the studies were connected with chief factors of a finite group. However, about the cover-avoiding property of subgroups for non-chief factor, there is no study up to now. The purpose of this paper is to build the theory. Let A be a subgroup of a finite group G and I pound: G (0) a (c) 1/2 G (1) a (c) 1/2 aEuro broken vertical bar a (c) 1/2 G (n) some subgroup series of G. Suppose that for each pair (K,H) such that K is a maximal subgroup of H and G (i-1) a (c) 1/2 K < H a (c) 1/2 G (i) for some i, either A a (c) H = A a (c) K or AH = AK. Then we say that A is I -embedded pound in G. In this paper, we study the finite groups with given systems of I -embedded pound subgroups. The basic properties of I -embedded pound subgroups are established and some new characterizations of some classes of finite groups are given and some known results are generalized.