Let G be a finite group and G p be a Sylow p-subgroup of G for a prime p in π(G), the set of all prime divisors of the order of G. The automiser A p (G) is defined to be the group N G (G p )/G p C G (G p ). We define the Sylow graph Γ A (G) of the group G, with set of vertices π(G), as follows: Two vertices p, q ∈ π(G) form an edge of Γ A (G) if either q ∈ π(A p (G)) or p ∈ π(A q (G)). The following result is obtained: * The second and third authors have been supported by Proyecto MTM2007-68010-C03-03, Ministerio de Educación y Ciencia and FEDER, Spain. The first author thanks the Universitat de València and the Universidad Politécnica de Valencia for their warm hospitality during the preparation of this paper.Theorem: Let G be a finite almost simple group. Then the graph Γ A (G) is connected and has diameter at most 5.We also show how this result can be applied to derive information on the structure of a group from the normalizers of its Sylow subgroups.
Let the group G = AB be a product of two π-decomposable sub-where π is a set of primes. The authors conjecture thatif π is a set of odd primes. In this paper it is proved that the conjecture is true if A and B are soluble. A similar result with certain additional restrictions holds in the case 2 ∈ π. Moreover, it is shown that the conjecture holds if O π (A) and O π (B) have coprime orders.
The main result in the paper states the following: Let π be a set of odd primes. Let the finite group G = AB be the product of a π -decomposable subgroup A = O π (A) × O π (A) and a π -subgroup B. Then O π (A) O π (G); equivalently the group G possesses Hall π -subgroups. In this case O π (A)B is a Hall π -subgroup of G. This result extends previous results of Berkovich (1966), Rowley (1977), Arad and Chillag (1981) and Kazarin (1980) where stronger hypotheses on the factors A and B of the group G were being considered. The results under consideration in the paper provide in particular criteria for the existence of non-trivial soluble normal subgroups for a factorized group G.
Two subgroups X and Y of a group G are said to be conditionallyi.e., X Y g is a subgroup of G. Using this permutability property new criteria for the product of finite supersoluble groups to be supersoluble are obtained and previous results are recovered. Also the behaviour of the supersoluble residual in products of finite groups is studied.
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