Communicated by M. Broué MSC:Primary: 20E26 a b s t r a c t A group is called extended residually finite (ERF) if every subgroup is closed in the profinite topology. The ERF-property is studied for nilpotent groups, soluble groups, locally finite groups and FC-groups. A complete characterization is given of FC-groups which are ERF.

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A well-known result of Feinberg and Shannon states that the tribonacci sequence can be detected by the so-called Pascal's pyramid. Here we will show that any tribonacci-like sequence can be obtained by the diagonals of the Feinberg's triangle associated to a suitable generalized Pascal's pyramid. The results also extend similar properties of Fibonacci-like sequence

We consider the well-known characterization of the Golden ratio as limit of the ratio of consecutive terms of the Fibonacci sequence, and we give an explanation of this property in the framework of the Difference Equations Theory. We show that the Golden ratio coincides with this limit not because it is the root with maximum modulus and multiplicity of the characteristic polynomial, but, from a more general point of view, because it is the root with maximum modulus and multiplicity of a restricted set of roots, which in this special case coincides with the two roots of the characteristic polynomial. This new perspective is the heart of the characterization of the limit of ratio of consecutive terms of all linear homogeneous recurrences with constant coefficients, without any assumption on the roots of the characteristic polynomial, which may be, in particular, also complex and not real.

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