We report on a partial solution of the conjecture that the class of finite solvable groups can be characterised by 2-variable identities. The proof requires pieces from number theory, algebraic geometry, singularity theory and computer algebra. The computations were carried out using the computer algebra system SINGULAR.

For any number field $K$ with non-elementary $3$-class group ${\mathrm{Cl}}_3\left(K\right)\simeq C_{3^e}\times C_3$, $e\ge 2$, the punctured capitulation type $\varkappa \left(K\right)$ of $K$ in its unramified cyclic cubic extensions $L_i$, $1\le i\le 4$, is an orbit under the action of $S_3\times S_3$. By means of Artin’s reciprocity law, the arithmetical invariant $\varkappa \left(K\right)$ is translated to the punctured transfer kernel type $\varkappa \left(G_2\right)$ of the automorphism group $G_2=\mathrm{Gal}({\mathrm{F}}_3^2\left(K\right)/K)$ of the second Hilbert $3$-class field of $K$. A classification of finite $3$-groups $G$ with low order and bicyclic commutator quotient $G/G^{\text{'}}\simeq C_{3^e}\times C_3$, $2\le e\le 6$, according to the algebraic invariant...

Various commutators and associators may be defined in one-sided loops. In this paper, we approximate and compare these objects in the left and right loop reducts of a Catalan loop. To within a certain order of approximation, they turn out to be quite symmetrical. Using the general analysis of commutators and associators, we investigate the structure of a specific Catalan loop which is non-commutative, but associative, that appears in the original number-theoretic application of Catalan loops.

In this short note, it is shown that if $A,B$ are $H$-connected transversals for a finite subgroup $H$ of an infinite group $G$ such that the index of $H$ in $G$ is at least 3 and $H\cap H^u\cap H^v=1$ whenever $u,v\in G\setminus H$ and $u{v}^{-1}\in G\setminus H$ then $A=B$ is a normal abelian subgroup of $G$.

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 $m$, $n$.