Structure Of Finite Groups Under Certain Arithmetical Conditions On The Length Of The Conjugacy Classes

How the arithmetical conditions on the length of the conjugacy classes of a finite groupinfluence its structure is considered in this paper. There are three sections in our resuts.In the first section, we talk about the influence of finite groups structure under theconjugacy classes lengths of elements of p-regular or prime power order are square-free (orcubic-free). Some main results as follows:Theorem 2.1.1 Let G be a non-abelian group and p the smallest prime divisor of|G|, P∈Sylp(G). If p2 does not divide |C| for each C∈Conp(G), then(1) G is a solvable p- nilpotent group;(2) If G is nilpotent, then |P : Op(G)| = 1 or p.Theorem 2.1.2 Let G be a solvable group and p the smallest prime divisor of |G|.If p3 does not divide |C| for each C∈Conp(G), G is unrelated with all [(Zp×Zp)]Zq types'Frobenius groups or 3 does not divide |G|, then G is p- nilpotent.Theorem 2.1.7 Let A be a normal subgroup and B a quasinormal subgroup of Gsuch that G = AB. Suppose that B is supersolvable and |xG| is square-free for every elementx of prime power order of A, then G is supersolvable.In the second section, the elements of a group G will be divided into two parts -πpart andπpart, then the relationship between the arithmetical conditions on the length ofthe conjugacy classes and the structure of finite groups is discussed. Some main results asfollows:Theorem 2.2.1 Let and every prime inπless than each one inπ∩π(G).If p2 does not divide |C| for each p∈πand C∈Con (G), then G isπ-supersolvable. Theorem 2.2.3 Let Zi(G) be ith in the ascending central series,π?π(G) and everyprime inπless than each one inπ∩π(G/Zi(G)). If p2 does not divide |C| for each p∈πand C∈Con(G), then G isπ-supersolvable.Theorem 2.2.9 Let Z(G) = 1 and the Conjugacy classes length of G exactly {1,for some distinct primes pi, where nj≥1, i, j = 1, 2,···s. then |G| =s and G is the direct product of groups G1, G2,···, Gn with the followingproperties:(1) (|Gi|, |Gj |) = 1 for i≠j;(2) The order of Gi is divisible by exactly two different primes and its Sylow subgroupsare abelian. In particular s is even.In the third section, inspired by the Deskins, Janko and Thompson theorem, we obtainsome sufficient condions of solvable groups, supersolvable groups, p-nilpotent groups andnilpotent groups by studying some arithmetical conditions on the length of conjugacy classesof the elments out of some maximal subgroups. Some main results as follows:Theorem 2.3.1 Let H be a nilpotent maximal subgroup of G and 4 don't divide |xG|for each x∈G \ H, then G is solvable.Theorem 2.3.2 Let H be a normal maximal subgroup of G and p the smallest primedivisor of |G|. If H is p- nilpotent and p2 does not divide |xG| for each x∈G\H, then G isp- nilpotent.Theorem 2.3.4 Let H be a nilpotent maximal subgroup of G. If |G : H| is a primeand |xG| square-free for each x∈G \ H, then G is supersolvable.