Browsing by Author "Russo, Francesco G"

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    A probabilistic approach to a classical result of ore
    (2021) Muhie, Seid Kassaw; Russo, Francesco G
    The subgroup commutativity degree sd(G) of a finite group G was introduced almost ten years ago and deals with the number of commuting subgroups in the subgroups lattice L(G) of G. The extremal case sd(G) = 1 detects a class of groups classified by Iwasawa in 1941 (in fact sd(G) represents a probabilistic measure which allows us to understand how far is G from the groups of Iwasawa). Among them we have sd(G) = 1 when L(G) is distributive, that is, when G is cyclic. The characterization of a cyclic group by the distributivity of its lattice of subgroups is due to a classical result of Ore in 1938. Therefore sd(G) is strongly related to structural properties of L(G). Here we introduce a new notion of probability gsd(G) in which two arbitrary sublattices S(G) and T(G) of L(G) are involved simultaneously. In case S(G) = T(G) = L(G), we find exactly sd(G). Upper and lower bounds in terms of gsd(G) and sd(G) are among our main contributions, when the condition S(G) = T(G) = L(G) is removed. Then we investigate the problem of counting the pairs of commuting subgroups via an appropriate graph. Looking at the literature, we noted that a similar problem motivated the permutability graph of non–normal subgroups ΓN (G) in 1995, that is, the graph where all proper non– normal subgroups of G form the vertex set of ΓN (G) and two vertices H and K are joined if HK = KH. The graph ΓN (G) has been recently generalized via the notion of permutability graph of subgroups Γ(G), extending the vertex set to all proper subgroups of G and keeping the same criterion to join two vertices. We use gsd(G), in order to introduce the non–permutability graph of subgroups ΓL(G) ; its vertices are now given by the set L(G) − CL(G)(L(G)), where CL(G)(L(G)) is the smallest sublattice of L(G) containing all permutable subgroups of G, and we join two vertices H, K of ΓL(G) if HK 6= KH. We finally study some classical invariants for ΓL(G) and find numerical relations between the number of edges of ΓL(G) and gsd(G).
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    Topics of entropy in locally compact abelian groups
    (2021) Waka, Olwethu; Russo, Francesco G
    The present MSc thesis discusses some notions of abelian group theory in connection with recent topics of topological entropy of locally compact abelian groups. It has been used the reference of [D. J. S. Robinson, A Course in the Theory of Groups, Springer, 1996, New York], which is a classical textbook in group theory. A list of exercises, relevant to our purposes, has been selected, in order to introduce some recent aspects of topological entropy of locally compact abelian groups. It is worth to mention that many of the exercises, which have been solved in the present thesis, are subject to technicalities which require the application of theorems of decomposition for abelian groups. Therefore the logic of the solutions allows us to describe the topological entropy in presence of an appropriate factorization.