Scriviamo $[x,y]=[x,{}_{1}y]$ ed $\left[\right[x,{}_{k}y],y]=[x,{}_{k+1}y]$. Cerchiamo gruppi $SL\left(2,q\right)$ con generatori $x,y$ tali che $[x,{}_{m}y]=x$ ed $[y,{}_{n}x]=y$ per alcuni numeri naturali $m$, $n$.

We show that there exists a finitely generated group of growth $\sim f$ for all functions $f:{ℝ}_{+}\to {ℝ}_{+}$ satisfying $f\left(2R\right)\le f{\left(R\right)}^2\le f\left({\eta }_{+}R\right)$ for all $R$ large enough and ${\eta }_{+}\approx 2.4675$ the positive root of $X^3-X^2-2X-4$. Set ${\alpha }_{-}=log2/log{\eta }_{+}\approx 0.7674$; then all functions that grow uniformly faster than $exp\left(R^{{\alpha }_{-}}\right)$ are realizable as the growth of a group.We also give a family of sum-contracting branched groups of growth $\sim exp\left(R^{\alpha }\right)$ for a dense set of $\alpha \in [{\alpha }_{-},1]$.

The structure of (generalized) soluble groups for which the set of all subnormal non-normal subgroups satisfies the maximal condition is described, taking as a model the known theory of groups in which normality is a transitive relation.

This paper deals with a rationality condition for groups. Let n be a fixed positive integer. Suppose every element g of the finite solvable group is conjugate to its nth power g n. Let p be a prime divisor of the order of the group. We conclude that the multiplicative order of n modulo p is small, or p is small.

This paper studies groups G whose all subgroups are either ascendant or self-normalizing. We characterize the structure of such G in case they are locally finite. If G is a hyperabelian group and has the property, we show that every subgroup of G is in fact ascendant provided G is locally nilpotent or non-periodic. We also restrict our study replacing ascendant subgroups by permutable subgroups, which of course are ascendant [Stonehewer S.E., Permutable subgroups of infinite groups, Math. Z., 1972,...

Our main result is that a locally graded group whose proper subgroups are Baer-by-Chernikov is itself Baer-by-Chernikov. We prove also that a locally (soluble-by-finite) group whose proper subgroups are Baer-by-(finite rank) is itself Baer-by-(finite rank) if either it is locally of finite rank but not locally finite or it has no infinite simple images.

If $𝒳$ is a class of groups, then a group $G$ is said to be minimal non $𝒳$-group if all its proper subgroups are in the class $𝒳$, but $G$ itself is not an $𝒳$-group. The main result of this note is that if $c\>0$ is an integer and if $G$ is a minimal non $\left(\mathrm{ℒℱ}\right)𝒩$ (respectively, $\left(\mathrm{ℒℱ}\right){𝒩}_c$)-group, then $G$ is a finitely generated perfect group which has no non-trivial finite factor and such that $G/Frat\left(G\right)$ is an infinite simple group; where $𝒩$ (respectively, ${𝒩}_c$, $\mathrm{ℒℱ}$) denotes the class of nilpotent (respectively, nilpotent of class at most $c$, locally...