Let $F$ be a field, $A$ be a vector space over $F$, $GL(F,A)$ be the group of all automorphisms of the vector space $A$. A subspace $B$ is called almost $G$-invariant, if ${dim}_F(B/{Core}_G\left(B\right))$ is finite. In the current article, we begin the study of those subgroups $G$ of $GL(F,A)$ for which every subspace of $A$ is almost $G$-invariant. More precisely, we consider the case when $G$ is a periodic group. We prove that in this case $A$ includes a $G$-invariant subspace $B$ of finite codimension whose subspaces are $G$-invariant.

Let F be a field, A be a vector space over F, GL(F, A) be the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dimF(BFG/B) is finite. A subspace B is called almost G-invariant, if dim F(B/Core G(B)) is finite. In the current article, we study linear groups G such that every subspace of A is either nearly G-invariant or almost G-invariant in the case when G is a soluble p-group where p = char F.

Fitting classes and injectors are discussed in the class of ${𝔊}_1$-groups. A necessary and sufficient condition for the existence of injectors is given; it is also shown that, when this condition holds, the injectors form a unique conjugacy class.

Let A be an R G-module, where R is an integral domain and G is a soluble group. Suppose that C G(A) = 1 and A/C A(G) is not a noetherian R-module. Let L nnd(G) be the family of all subgroups H of G such that A/C A(H) is not a noetherian R-module. In this paper we study the structure of those G for which L nnd(G) satisfies the maximal condition.

We show that a locally graded group with a finite number m of non-(nilpotent of class at most n) subgroups is (soluble of class at most [log₂n] + m + 3)-by-(finite of order ≤ m!). We also show that the derived length of a soluble group with a finite number m of non-(nilpotent of class at most n) subgroups is at most [log₂ n] + m + 1.

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.

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.