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IN questo lavoro ci si propone insomma un percorso di gioco, osservazione, scoperta, definizione, formalizzazione, verifica che, con linguaggio e metodi logico-matematici affronti e sviluppi problematiche significative per il mondo reale e per il cittadino. Questo Laboratorio vuole avvicinare i partecipanti al progetto al concetto di immagine digitale dal punto di vista di un oggetto che vive e viene elaborato attraverso modelli più o meno complicati della Matematica

si studiano gli automorfismi dei gruppi residualmente finiti mediante l'analisi della loro azione sulle immagini caratteristiche finite.

si perviene a caratterizzare, in vari termini e sotto deboli ipotesi di risolubilita', i gruppi in cui ogni sottogruppo e' intersezione di sottogruppi normali o massimali ed in particolare i gruppi risolubili nei quali i sottogruppi non-normali sono intersezione di sottogruppiu massimali.

si studiano gruppi in cui sottogruppi appartenenti a rilevanti sistemi di sottogruppi sono M-sottogruppi, dove per M-sottogruppo si intende un sottogruppo che e' intersezione di sottogruppi massimali.

‎‎Let $G$ be a group and $p$ be an endomorphism of $G$‎. ‎A subgroup $H$ of $G$ is called $p$-inert if $H^pcap H$ has finite index in the image $H^p$‎. ‎The subgroups that are $p$-inert for all inner automorphisms of $G$ are widely known and studied in the literature‎, ‎under the name inert subgroups‎. ‎The related notion of inertial endomorphism‎, ‎namely an endomorphism $p$ such that all subgroups of $G$ are $p$-inert‎, ‎was introduced in cite{DR1} and thoroughly studied in cite{DR2,DR4}‎. ‎The ``dual‎" ‎notion of fully inert subgroup‎, ‎namely a subgroup that is $p$-inert for all endomorphisms of an abelian group $A$‎, ‎was introduced in cite{DGSV} and further studied in cite{Ch+‎, ‎DSZ,GSZ}‎. ‎The goal of this paper is to give an overview of up-to-date known results‎, ‎as well as some new ones‎, ‎and show how some applications of the concept of inert subgroup fit in the same picture even if they arise in different areas of algebra‎. ‎We survey on classical and recent results on groups whose inner automorphisms are inertial‎. ‎Moreover‎, ‎we show how‎ ‎inert subgroups naturally appear in the realm of locally compact topological groups or locally linearly compact topological vector spaces‎, ‎and can be helpful for the computation of the algebraic entropy of continuous endomorphisms‎.

A subgroup H of a group G is called inert if‎, ‎for each g∈G ‎, ‎the index of H∩H g in H is finite‎. ‎We give a classification ‎of soluble-by-finite groups G in which subnormal subgroups are inert in the cases where G has no nontrivial torsion normal subgroups or G ‎ ‎is finitely generated‎.

A group G is a cn -group if for each subgroup H of G there exists a normal subgroup N of G such that the index | H N : ( H ∩ N ) | is finite. The class of cn -groups contains properly the classes of core-finite groups and that of groups in which each subgroup has finite index in a normal subgroup. In the present paper it is shown that a cn -group whose periodic images are locally finite is finite-by-abelian-by-finite. Such groups are then described into some details by considering automorphisms of abelian groups. Finally, it is shown that if G is a locally graded group with the property that the above index is bounded independently of H, then G is finite-by-abelian-by-finite.

ADV Perspectives in Group Theory – an open space –

We study the group IAut ⁡ ( A ) ${\operatorname{IAut}(A)}$ generated by the inertial automorphisms of an abelian group A, that is, automorphisms γ ${{\gamma}}$ with the property that each subgroup H of A has finite index in the subgroup generated by H and H ⁢ γ ${H{\gamma}}$ . Clearly, IAut ⁡ ( A ) ${\operatorname{IAut}(A)}$ contains the group FAut ⁡ ( A ) ${\operatorname{FAut}(A)}$ of finitary automorphisms of A, which is known to be locally finite. In a previous paper, we showed that IAut ⁡ ( A ) ${\operatorname{IAut}(A)}$ is (locally finite)-by-abelian. In this paper, we show that IAut ⁡ ( A ) ${\operatorname{IAut}(A)}$ is also metabelian-by-(locally finite). More precisely, IAut ⁡ ( A ) ${\operatorname{IAut}(A)}$ has a normal subgroup Γ ${{\Gamma}}$ such that IAut ⁡ ( A ) / Γ ${\operatorname{IAut}(A)/{\Gamma}}$ is locally finite and the derived subgroup Γ ′ ${{\Gamma}^{\prime}}$ is an abelian periodic subgroup all of whose subgroups are normal in Γ ${{\Gamma}}$ . In the case when A is periodic, IAut ⁡ ( A ) ${\operatorname{IAut}(A)}$ turns out to be abelian-by-(locally finite) indeed, while in the general case it is not even (locally nilpotent)-by-(locally finite). Moreover, we provide further details about the structure of IAut ⁡ ( A ) ${\operatorname{IAut}(A)}$ .

We show that if a group G has a finite normal subgroup L such that G / L is hypercentral, then the index of the (upper) hypercenter of G is at most | Aut ( L ) | ⋅ | L | . It follows an explicit bound for | G / Z 2 m ( G ) | in terms of d = | γ m + 1 ( G ) | and independent of m ∈ N , provided d is finite. This completes recent results generalizing classical theorems by R. Baer and P. Hall. Then we extend other results in the literature by applying our results to groups of automorphisms acting in a restricted way on an ascending normal series of a group G.

In this paper we propose a reflection on the use of axiomatic set theory as a fundamental tool to address the foundational issues of mathematics. In particular, we focus on the key concept of infinity, indeed the strongest kind of infinity which may be conceived (what is referred to as “the Absolute” by Cantor), as we aim to show how the point of view offered by a specific set-theoretical framework allows us to deal with such a paradoxical notion in a completely safe manner. For this purpose, we shall introduce NBG set theory and discuss its consistency. We assume the reader is familiar with ZFC set theory (see for example [2] or [3]). Mathematics Subject Classification: 03E99

We consider groups Γ of automorphisms of a groupG acting by means of power automorphisms on the factors of a normal series inG with lengthm. We show that [G, Γ] is nilpotent with class at mostm and that this bound is best possible. Moreover, such a Γ is parasoluble with paraheight at most 1/2m(m+3)+1, provided Γ′ is periodic. We give best possible bound in the case where the series is a central one.

Si caratterizzano gli automorfismi di un gruppo abeliano che trasformano sottogruppi H in sottogruppi commensurabili K tali che H intesecato K ha indice finito sia in H che in K.