Some Basic Properties Of Groups Influenced By Their Maximal Subgroups

?One of the most important and effective ways is that the structure of group is established by maximal subgroup.In document [20],the author studied the structure of the finite non-simple group which possesses a simple maximal subgroup.Influenced by document [20],In this paper we discuss simplicity of groups mainly by studing groups which possess simple maximal subgroups.In addition,we also study solubility of groups by their some especil maximal subgroups.lt is worth noticing that the group studied in this paper is an arbitary group,finite or infinite.We get the following theorems:About simplicityTheorem 3.1. Let the maximal subgroups of group G be all simple.If there is a nonnormal maximal subgroup of G which possesses property p,then G is simple.Theorem 3.2. Let G be finitely generated.If the maximal subgroups of group G are all nonnormal, and are simple,then G is simple.Theorem 3.3. Let the maximal subgroups of group G be all nonnormal, and be either simple or nilpotent, if both of the two cases are existent,and one of maximal subgroups of G is finite.then G is simple.Theorem 3.4. Let the maximal subgroups of group G be all nonnormal, and be either simple or nilpotent, if one of maximal subgroups is nilpotent and finitely generated and other one of maximal subgroups is simple and torsion, then G is simple.Theorem 3.5. Let the maximal subgroups of group G be all nonnormal.and be either simple or nilpotent, if there exists a maximal subgroup M be nilpotent and maximal subgroup R be simple and satisfy R(?)M per M, then G is simple.Remark: The Tarski group is generated by two elements and infinitely simple.The maximal subgroups of the Tarski group are prime order.Corollary 3.3. The order of any maximal subgroup of group G is prime if and only if G is an Tarski group or a group of order pq.Theorem 3.6. Let H be a. nonormal and simple and maximal subgroup of group G , if there exist nontrival element h₁ and h₂ in H,such that h₁ and h₂ are conjugate each other in G but not in H,then G is simple.Theorem 3.7 let H be a nonormal,simple,and maximal subgroup of group G, if |G:H|≤min{|g^G||1≠g∈G},then G is simple.About solubilityTheorem 4.1 Let the maximal subgroups of group G be all normal,if there are only n maximal subgroups for group G,then G is finite.and certainly,G is also nilpotent.Theorem 4.2 Let G = 1,g₂,…,g_n>then every maximal subgroup of G is a normal subgroup and the elements of G which are conjugate each other in G are also conjugate in the maximal subgroups which they are contained if and only if G is an Abel group.Theorem 4.3 If group G is finitely generated,and there are at least two maximal subgroups in G,then G is soluble if and only if every maximal subgroup of G is soluble and Frat(G) is not a maximal normal subgroup of G.Theorem 4.4 Let G be a finite group and every maximal subgroup of G be soluble,if N≤G and there is a prime p ,such that |G|_p = |N|_p,then G is soluble.Theorem 4.5 Let every maximal subgroup of group G be nilpotent,if there is a maximal subgroup of G whose index is finite,then G is soluble.Theorem 5.4 LetÏ€be a set of odd numbers and G be aÏ€-group,if there is a soluble subgroup of G whose index is finite.then G is soluble .Theorem 5.5 Suppose p, q are distinct primes,LetÏ€= {p, q},if G is aÏ€-group and there is a soluble subgroup of G whose index is finite .then G is soluble.