Un sottogruppo $H$ di un gruppo $G$ si dice «almost normal» se ha soltanto un numero finito di coniugati in $G$, e ovviamente l'insieme $an\left(G\right)$ costituito dai sottogruppi almost normal di $G$ è un sottoreticolo del reticolo $\mathcal{L}\left(G\right)$ di tutti i sottogruppi di $G$. In questo articolo vengono studiati gli isomorfismi tra reticoli di sottogruppi almost normal, provando in particolare che se $G$ è un gruppo supersolubile e $\overline{G}$ è un gruppo FC-risolubile tale che i reticoli $an\left(G\right)$ e $an\left(\overline{G}\right)$ sono isomorfi, allora anche $\overline{G}$ è supersolubile, e...

In this work it is shown that a locally graded minimal non CC-group G has an epimorphic image which is a minimal non FC-group and there is no element in G whose centralizer is nilpotent-by-Chernikov. Furthermore Theorem 3 shows that in a locally nilpotent p-group which is a minimal non FC-group, the hypercentral and hypocentral lengths of proper subgroups are bounded.

A group $G$ is said to be a PC-group, if $G/C_G\left(x^G\right)$ is a polycyclic-by-finite group for all $x\in G$. A minimal non-PC-group is a group which is not a PC-group but all of whose proper subgroups are PC-groups. Our main result is that a minimal non-PC-group having a non-trivial finite factor group is a finite cyclic extension of a divisible abelian group of finite rank.

We show that a finite nonabelian characteristically simple group $G$ satisfies $n=\left|\pi \right(G\left)\right|+2$ if and only if $G\cong A_5$, where $n$ is the number of isomorphism classes of derived subgroups of $G$ and $\pi \left(G\right)$ is the set of prime divisors of the group $G$. Also, we give a negative answer to a question raised in M. Zarrin (2014).

A subgroup H of a group G is inert if |H: H ∩ H g| is finite for all g ∈ G and a group G is totally inert if every subgroup H of G is inert. We investigate the structure of minimal normal subgroups of totally inert groups and show that infinite locally graded simple groups cannot be totally inert.

This article is dedicated to some criteria of generalized nilpotency involving pronormality and abnormality. Also new results on groups, in which abnormality is a transitive relation, have been obtained.

In questo lavoro studiamo i non CC-gruppi $G$ monolitici con tutti i quozienti propri CC-gruppi, che hanno sottogruppi abeliani normali non banali.

We characterize the solvable groups without infinite properly ascending chains of non-BFC subgroups and prove that a non-BFC group with a descending chain whose factors are finite or abelian is a Cernikov group or has an infinite properly descending chain of non-BFC subgroups.

Let $G$ be a group and $n$ an integer $\ge 2$. We say that $G$ has the $n$-permutation property $(G\in P_n)$ if, for any elements $x_1,x_2,\mathrm{\cdots },x_n$ in $G$, there exists some permutation $\sigma$ of $\{1,2,\mathrm{\cdots },n\}$, $\sigma \ne id.$ such that $x_1,x_2,\mathrm{\cdots },x_n=x_{\sigma (1)},x_{\sigma (2)},\mathrm{\cdots },x_{\sigma (n)}$. We prouve that every group $G\in P_n$ is an FC-nilpotent group of class $\le n-1$, and that a finitely generated group has the $n$-permutation property (for some $n$) if, and only if, it is abelian by finite. We prouve also that a group $G\in P_3$ if, and only if, its derived subgroup has order at most 2.