| It is important to study the structures of finite groups by combining some subgroups with some properties in the researches on finite groups. Many scholars have studied these respects and given many important results, for example, the Huppert's famous theorem, namely, a finite group G is supersolvable if and only if every maximal subgroup of G has prime index; a finite group G is nilpotent if and only if every maximal subgroup of G is normal in G; a finite group G is solvable if and only if every maximal subgroup of G is c-normal in G (See it in [70]); etc. In this paper, we study the properties of normality, θ-pairs and c-section of subgroups in finite groups, with which we describe the structure of some finite groups and get some meaningful results. We divide this paper into four chapters. The main of the paper is the following:In chapter 1, we introduce some symbols and basic concepts that we usually use in the paper.In chapter 2, we define and study conceptions: πSCAP-subgroup and nc-subgroup. With these concepts we study the influence of subgroups on finite groups and obtain some results on (π-)solvability, p-nilpotence, or (π-)supersolvability. And some of these results are generalization of some known results. In the end of this chapter, we introduce some concepts on the normal properties of subgroups and relations on them.In chapter 3, we mainly study the properties of θ-pairs for subgroups and get some sufficient or sufficient and necessary conditions on solvability or nilpotence of finite groups. In §3.3, we study the number of θ-pairs for maximal subgroups and solved a question in [6] that was posed by A. R. Ashra(?) and R. Soleimani, namely, " Is there a non-abelian finite group with exactly n θ-pair? ". The answer to this question is that there is not a non-abelian finite group with exactly 4 θ—pair, but a non-abelian finite group with exactly n θ-pair for n > 4. In §3.4, we generalize θ-pairs for maximal subgroups to θ-pairs for subgroup and get some sufficient or sufficient and necessary conditions on solvability or nilpotence of finite groups.In chapter 4, we study c-section of maximal subgroups, and get some necessary and sufficient conditions on solvability of finite groups. Especially, we have proved that if Sec(M) is supersolvable for every maximal subgroup M of G, then any composition factor of G is isomorphic to L2(p) or Zq, where p and q are primes, and p ≡ ±1(mod 8). This answers the question "For every maximal subgroup M of a group G assume that Sec(M) is supersolvable. Is G solvable?", which is posed by Y. Wang and S. Li in [72]. At the end, we study some supplements with special properties and get two sufficient conditions on solvability of finite groups.
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