Search results for "NUMBER"
showing 10 items of 3939 documents
Abelian Sylow subgroups in a finite group, II
2015
Abstract Let p ≠ 3 , 5 be a prime. We prove that Sylow p-subgroups of a finite group G are abelian if and only if the class sizes of the p-elements of G are all coprime to p. This gives a solution to a problem posed by R. Brauer in 1956 (for p ≠ 3 , 5 ).
ON SYLOW NORMALIZERS OF FINITE GROUPS
2013
[EN] The paper considers the influence of Sylow normalizers, i.e. normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup- closed saturated formations satisfying the following property: a group belongs to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.
On second minimal subgroups of Sylow subgroups of finite groups
2011
A subgroup H of a finite group G is a partial CAP-subgroup of G if there is a chief series of G such that H either covers or avoids its chief factors. Partial cover and avoidance property has turned out to be very useful to clear up the group structure. In this paper, finite groups in which the second minimal subgroups of their Sylow p-subgroups, p a fixed prime, are partial CAP-subgroups are completely classified.
The Fitting Subgroup and Some Injectors of Radical Locally Finite Groups with min-pfor Allp
2003
Abstract This work was intended as an attempt to continue the study of the class ℬ of generalised nilpotent groups started in a previous paper. We present some results concerning the Fitting subgroup and the ℬ-injectors of a radical locally finite group satisfying min-p for all p.
A Gaschütz–Lubeseder Type Theorem in a Class of Locally Finite Groups
1999
The aim of this paper is to present a Gaschutz–Lubeseder type theorem in the class cL of all radical locally finite groups satisfying min−p for all primes p. Notice that these groups are countable and co-Hopfian by [1, (5.4.8)]. In retrospect, the theory of saturated formations of finite soluble groups began with the results of Gaschutz [3] in 1963. He introduced the concept of “covering subgroup” as a generalization of Sylow and Hall subgroups. These covering subgroups have many of the properties of Sylow and Hall subgroups other than the arithmetic ones. The main idea of Gaschutz’s work was concerned with group theoretical classes having the same properties. He defined a formation F to be…
A note on Sylow permutable subgroups of infinite groups
2014
Abstract A subgroup A of a periodic group G is said to be Sylow permutable, or S-permutable, subgroup of G if A P = P A for all Sylow subgroups P of G. The aim of this paper is to establish the local nilpotency of the section A G / Core G ( A ) for an S-permutable subgroup A of a locally finite group G.
On some classes of supersoluble groups
2007
[EN] Finite groups G for which for every subgroup H and for all primes q dividing the index |G:H| there exists a subgroup K of G such that H is contained in K and |K:H|=q are called Y-groups. Groups in which subnormal subgroups permute with all Sylow subgroups are called PST-groups. In this paper a local version of the Y-property leading to a local characterisation of Y-groups, from which the classical characterisation emerges, is introduced. The relationship between PST-groups and Y-groups is also analysed.
Finite groups with all minimal subgroups solitary
2016
We give a complete classification of the finite groups with a unique subgroup of order p for each prime p dividing its order. All the groups considered in this paper will be finite. One of the most fruitful lines in the research in abstract group theory during the last years has been the study of groups in which the members of a certain family of subgroups satisfy a certain subgroup embedding property. The family of the subgroups of prime order (also called minimal subgroups) has attracted the interest of many mathematicians. For example, a well-known result of Itˆo (see [8, Kapitel III, Satz 5.3; 9]) states that a group of odd order with all minimal subgroups in the center is nilpotent. Th…
On the Frattini subgroup of a finite group
2016
We study the class of finite groups $G$ satisfying $\Phi (G/N)= \Phi(G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe to all finite groups and answer in the affirmative a long-standing question of Christensen whether the class of finite groups which possess complements for each of their normal subgroups is subnormally closed.
A Complete, Exact and Efficient Implementation for Computing the Edge-Adjacency Graph of an Arrangement of Quadrics
2011
International audience; We present a complete, exact and efficient implementation to compute the edge-adjacency graph of an arrangement of quadrics, i.e. surfaces of algebraic degree 2. This is a major step towards the computation of the full 3D arrangement. We enhanced an implementation for an exact parameterization of the intersection curves of two quadrics, such that we can compute the exact parameter value for intersection points and from that the edge-adjacency graph of the arrangement. Our implementation is complete in the sense that it can handle all kinds of inputs including all degenerate ones, i.e. singularities or tangential intersection points. It is exact in that it always comp…