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...

The group $Au{t}_{sn}G$ of all automorphisms leaving invariant every subnormal subgroup of the group $G$ is studied. In particular it is proved that $Au{t}_{sn}G$ is metabelian if $G$ is soluble, and that $Au{t}_{sn}G$ is either finite or abelian if $G$ is polycyclic.

Using a lemma on subnormal subgroups, the problem of nilpotency of multiplication groups and inner permutation groups of centrally nilpotent loops is discussed.

An algebraic structure is said to be congruence permutable if its arbitrary congruences $\alpha$ and $\beta$ satisfy the equation $\alpha \circ \beta =\beta \circ \alpha$, where $\circ$ denotes the usual composition of binary relations. To an arbitrary $G$-set $X$ satisfying $G\cap X=\varnothing$, we assign a semigroup $(G,X,0)$ on the base set $G\cup X\cup \left\{0\right\}$ containing a zero element $0\notin G\cup X$, and examine the connection between the congruence permutability of the $G$-set $X$ and the semigroup $(G,X,0)$.

In questo lavoro si studiano i gruppi ${\text{Aut}}_{sn}\left(G\right)$, ${\text{Aut}}_d\left(G\right)$, ${\text{Aut}}_{\chi }\left(G\right)$ degli automorfismi di un gruppo $G$ che fissano — come insiemi — tutti i sottogruppi di $G$ che risultano essere rispettivamente subnormali, subnormali di difetto al più $d$, oppure che sono compresi tra un sottogruppo caratteristico ed il suo derivato. Si danno condizioni sufficienti affinché tali gruppi siano parasolubili di para-altezza al più 2 o 3. Si generalizzano così risultati da [4], [7], [8], [10].

2000 Mathematics Subject Classification: 20F16, 20E15.Groups in which every contranormal subgroup is normally complemented has been considered. The description of such groups G with the condition Max-n and such groups having an abelian nilpotent residual satisfying Min-G have been obtained.

A subgroup $M$ of a group $G$ is nearly maximal if the index $|G:M|$ is infinite but every subgroup of $G$ properly containing $M$ has finite index, and the group $G$ is called nearly $IM$ if all its subgroups of infinite index are intersections of nearly maximal subgroups. It is proved that an infinite (generalized) soluble group is nearly $IM$ if and only if it is either cyclic or dihedral.

It is proved that a soluble residually finite minimax group is finite-by-nilpotent if and only if it has only finitely many maximal subgroups which are not normal.

The current article considers some infinite groups whose finitely generated subgroups are either permutable or pronormal. A group $G$ is called a generalized radical, if $G$ has an ascending series whose factors are locally nilpotent or locally finite. The class of locally generalized radical groups is quite wide. For instance, it includes all locally finite, locally soluble, and almost locally soluble groups. The main result of this paper is the followingTheorem. Let $G$ be a locally generalized radical...