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

Given a set r of permutations of an n-set, let G be the group of permutations generated by r. I f p is any prime, it is known that a Sylow p-subgroup P of G can be found in polynomial time. We show that the normalizer of P can also be found in polynomial time. In particular, given two Sylow p-subgroups of G, all elements conjugating one to the other can be found (as a coset of the normalizer of...

‎Suppose G is a group of order p^2q^2 where p>q are prime numbers and suppose P and Q are Sylow p-subgroups and Sylow q-subgroups of G, ‎respectively‎. ‎In this paper‎, ‎we show that up to isomorphism‎, ‎there are four groups of order p^2q^2 when Q and P are cyclic‎, ‎three groups when Q is a cyclic and P is an elementary ablian group‎, ‎p^2+3p/2+7 groups when Q is an elementary ablian group an...

Let G be a finite group. We prove that if the number of Sylow 3-subgroups is at most 7 and 5-subgroups 1455, then solvable. This strong form recent conjecture Robati.

Alperin [1] has recently introduced a fundamental method for conjugating from one p-Sylow subgroup Q of a finite group G to a second p-Sylow subgroup P in a series of steps. The importance of this is that it provides information concerning the conjugacy in G of subsets of P, that is, fusion of subsets of P. In certain situations it is possible to extend these results to the case of conjugate p-...

A finite group G is said to be a POS-group if for each x in G the cardinality of the set {y in G | o(y) = o(x)} is a divisor of the order of G. In this paper we study the structure of POS-groups with some cyclic Sylow subgroups.

Solution. The 11-Sylow subgroup is Z/11Z; the 5-Sylow subgroup is Z/5Z. By Sylow’s theorem, the 11-Sylow subgroup is normal. Hence, the group is a semi-direct product of its 5 and 11-Sylow subgroups. Since Aut(Z/11Z) = Z/10Z has a unique subgroup of order 5, there are up to isomorphism exactly two groups of order 55: the abelian group Z/55Z and the group with presentation ⟨x, y|x = y = 1, xyx−1...

Given a set r of permutations of an n-set, let G be the group of permutations generated by f. If p is a prime, a Sylow p-subgroup of G is a subgroup whose order is the largest power of p dividing IGI. For more than 100 years it has been known that a Sylow p-subgroup exists, and that for any two Sylow p-subgroups PI, P, of G there is an element go G such that Pz = g-‘PI g. We present polynomial-...