A subgroup H of a group G is called ss-quasinormal in G if there is a subgroup B of G such that G = HB and H permutes with every Sylow subgroup of B; H is called weakly s-permutable in G if there is a subnormal subgroup T of G such that G = HT and H ∩ T ≤ HsG, where HsG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We fix in every non-cyclic Sylow subgr...

suppose that  is a finite group. then the set of all prime divisors of  is denoted by  and the set of element orders of  is denoted by . suppose that . then the number of elements of order  in  is denoted by  and the sizes of the set of elements with the same order is denoted by ; that is, . in this paper, we prove that if  is a group such that , where , then . here  denotes the family of suzuk...

A subgroup H is said to be s-permutable in a group G, if HP = PH holds for every Sylow subgroup P of G. If there exists a subgroup B of G such that HB = G and H permutes with every Sylow subgroup of B, then H is said to be SS-quasinormal in G. In this paper, we say that H is a weakly SS-quasinormal subgroup of G, if there is a normal subgroup T of G such that HT is s-permutable and H ∩ T is SS-...

Let H be a subgroup of a finite group G. We say that H is SS-semipermutable in Gif H has a supplement K in G such that H permutes with every Sylow subgroup X of Kwith (jXj; jHj) = 1. In this paper, the Structure of SS-semipermutable subgroups, and finitegroups in which SS-semipermutability is a transitive relation are described. It is shown thata finite solvable group G is a PST-group if and on...

In the Sylow theorems f we learn that if the order of a group 2Í is divisible hj pa (p a prime integer) and not by jo*+1, then 31 contains one and only one set of conjugate subgroups of order pa, and any subgroup of 21 whose order is a power of p is a subgroup of some member of this set of conjugate subgroups of 2Í. These conjugate subgroups may be called the Sylow subgroups of 21. It will be o...

In this paper, we prove the p-nilpotency of a finite group with assumption that some subgroups of Sylow subgroup are weakly s-semipermutable subgroups in the normalizer of Sylow subgroups. Our results unify and generalize some earlier results. Mathematics Subject Classification: 20D10, 20D15

We explicitly determine all of the transitive groups of degree p2, p a prime, whose Sylow p-subgroup is not isomorphic to the wreath product Z p Z p . Furthermore, we provide a general description of the transitive groups of degree p2 whose Sylow p-subgroup is isomorphic to Z p Z p , and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objec...

In finite group theory, one of the most influential ideas is the notion of Sylow p-subgroup. It was used to classify many groups of finite order. Despite original intentions to use it with finite group, it was extended to study Lie group, which inspires recent development in p-compact group. Here, we will discuss some aspects of Sylow’s Subgroup Theorem. In particular, the focus is on a finite ...

The third Conway group Co3 is one of the twenty-six sporadic finite simple groups. The cohomology of its Sylow 2-subgroup S is computed, an important step in calculating the mod 2 cohomology of Co3. The spectral sequence for the central extension of S is described and collapses at the sixth page. Generators are described in terms of the Evens norm or transfers from subgroups. The central quotie...