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For a non-empty class of groups $\cal L$, a finite group $G = AB$ is said to be an $\cal L$-connected product of the subgroups $A$ and $B$ if $\langle a, b\rangle \in \cal L$ for all $a \in A$ and $b \in B$. In a previous paper, we prove that for such a product, when $\cal L = \cal S$ is the class of finite soluble groups, then $[A,B]$ is soluble. This generalizes the theorem of Thompson which states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper our result is applied to extend to finite groups previous research in the soluble universe. In particular, we characterize connected products for relevant classes of groups; among others the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Also we give local descriptions of relevant subgroups of finite groups.

Let G be a finite group and Gp be a Sylow p-subgroup of G for a prime p in pi(G), the set of all prime divisors of the order of G. The automiser Ap(G) is defined to be the group NG(Gp)/GpCG(Gp). We define the Sylow graph gamma A(G) of the group G, with set of vertices pi(G), as follows: Two vertices p, q ¿ ¿(G) form an edge of ¿A(G) if either q ¿ ¿(Ap(G)) or p ¿ ¿(Aq(G)). The following result is obtained: Theorem: Let G be a finite almost simple group. Then the graph ¿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., The second and third authors have been supported by Proyecto MTM2007-68010-C03-03 and Proyecto MTM2010-19938-C03-02, Ministerio de Educacion y Ciencia and FEDER, Spain.