Let G be a finite p-soluble group, and P a SYLOW p-SUBGROUP of G. It is proved that if all elements of P of order p (or of order £ 4 for p=2) are contained in the k-th term of the upper central series of P, then the p-length of G is at most 2m+1, where m is the greatest integer such that pm-pm-1£k, and the exponent of the image of P in G=Op; p (G) is at most pm. It is also proved that if P is a powerful p-group, then the p-length of G is equal to 1.

Let G=SL2 (pf) be a special linear group and P be a SYLOW 2-SUBGROUP of G, where p is a prime and f is a positive integer such that pf>3. By NG (P) we denote the normalizer of P in G. In this paper, we show that NG (P) is nilpotent (or 2-nilpotent, or supersolvable) if and only if p2f º1 (mod 16).

In this paper, groups with trivial intersection between Frattini and derived sub-groups are considered. First, some structural properties of these groups are given in an important special case. Then, some family invariants of each n-isoclinism family of such groups are stated. In particular, an explicit bound for the order of each center factor group in terms of the order of its derived SUBGROUP is also provided.

Let G be a , nite group in which every SYLOW SUBGROUP is seminormal or abnormal. We prove that G has a SYLOW tower. We establish that if a group has a maximal SUBGROUP with SYLOW SUBGROUPs under the same conditions, then this group is soluble.

1. Introduction: There is a long-standing conjecture attributed to I. Schur that if G is a , nite group with Schur multiplier M(G) then the exponent of M(G) divides the exponent of G. It is easy to show that this is true for groups G of exponent 2 or exponent 3, but it has been known since 1974 that the conjecture fails for exponent 4. Bayes, Kautsky and Wamsley [1] give an example of a group G of order 2 with exponent 4, where M(G) has exponent 8. (Bayes, Kautsky and Wamsley are heros of the early days of computing with , nite p-groups. ) However the truth or otherwise of this conjecture has remained open up till now for groups of odd exponent, and in particular it has remained open for groups of exponent 5 and exponent 9. For a survey article on Schur's conjecture see Thomas [6]...

Generalizing the concept of quasinormality, a SUBGROUP H of a group G is said to be 4quasinormal in G if, for all cyclic SUBGROUPs K of G, ⟨ H; K ⟩ = HKHK. An intermediate concept would be 3-quasinormality, but in finite p-groups-our main concern-this is equivalent to quasinormality. Quasinormal SUBGROUPs have many interesting properties and it has been shown that some of them can be extended to 4-quasinormal SUBGROUPs, particularly in finite p-groups. However, even in the smallest case, when H is a 4-quasinormal SUBGROUP of order p in a finite p-group G, precisely how H is embedded in G is not immediately obvious. Here we consider one of these questions regarding the commutator SUBGROUP [H; G].

Following P. Hall a soluble group whose SYLOW SUBGROUPs are all abelian is called A-group. The purpose of this article is to give a new and shorter proof for a criterion on the capability of A-groups of order p2q, where p and q are distinct primes. Subsequently we give a sufficient condition for n-capability of groups having the property that their center and derived SUBGROUPs have trivial intersection, like the groups with trivial Frattini SUBGROUP and A-groups. An interesting necessary and sufficient condition for capability of the A-groups of square free order will be also given.

Denote by $ G $ a finite group, by  $ {\rm hsn}(G) $ the harmonic mean SYLOW number (eliminating the SYLOW numbers that are one) in $G$ and by    $ {\rm gsn}(G) $ the geometric mean SYLOW number (eliminating the SYLOW numbers that are one) in $G$. In this paper, we prove that if either $ {\rm hsn}(G)<45/7 $ or  $ {\rm gsn}(G)< \sqrt[3]{300} $, then $G$ is solvable. Also, we show that if either $ {\rm hsn}(G)<24/7 $ or  $ {\rm gsn}(G)<\sqrt{12} $, then $G$ is supersolvable.