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.

Abstract We prove that the number of conjugacy classes a finite group G consisting elements odd order, is larger than or equal to for normaliser Sylow 2-subgroup . This predicted by Alperin Weight Conjecture.

We introduce a strong form of Oliver’s p-group conjecture and derive a reformulation in terms of the modular representation theory of a quotient group. The Sylow p-subgroups of the symmetric group Sn and of the general linear group GLn(Fq) satisfy both the strong conjecture and its reformulation.

Even Type Conjecture. Let G be a simple group of finite Morley rank of even type. Then G is a Chevalley group over an algebraically closed field of characteristic two. See [16] for an informal introduction to the subject, [1] for a recent survey of the classification programme, and [17] for general background on groups of finite Morley rank. An infinite simple group G of finite Morley rank whos...

In this paper, we construct a group with three real irreducible characters whose Sylow 2-subgroup is an iterated central extension of a Suzuki 2-group. This answers a question raised by Moretó and Navarro who asked whether such a group exists. MSC primary: 20C15

A finite p-group S is said to be of supersoluble type if every fusion system over supersoluble. The main aim this paper characterise the p-groups type. Abelian and metacyclic are completely described. Furthermore, we show that Sylow p-subgroups a simple group must cyclic.

The novel notion of rigid commutators is introduced to determine the sequence logarithms indices a certain normalizer chain in Sylow 2-subgroup symmetric group on 2^n letters. terms this are proved be those partial sums partitions an integer into at least two distinct parts, that relates famous Euler's partition theorem.

In this paper, we find a strong new restriction on the structure of CI-groups. We show that, if R is generalised dihedral group and CI-group, then for every odd prime p Sylow p-subgroup has order p, or 9. Consequently, any CI-group with quotient same restriction, that

We enumerate and describe the Sylow p-groups of the group of polynomial permutations of the integers mod p . MSC 2000: primary 20D20, secondary 11T06, 13M10, 11C08, 13F20, 20E18.