Algorithmic Problems Related to the Direct Prodict of Groups

Nataša Božović, Sava Krstić

Algorithmic unrecognizability of every group property from classes $\Cal K'_1$ and $\Cal K'_2$ of properties (of $fp$ groups) related to the direct product of groups, is proved. $\Cal K'_1$ is the class of all properties of the form ``being a direct product of groups with Markov properties''. $\Cal K_2$ is the class of properties $P=S\cup T$, where $S$ is a property of universal $fp$ groups only and $T$ is a property of nonuniversal groups, such that there exists a positive integer $m$ for which $$ (\forall G\in S)d(G)\not=m\vee(\forall G\in T)d(G)\not=m, $$ where $d(G)=\sup\{k\mid(\exists H_1,\ldots, H_k\not=\{1\})(G\cong H_1\times\cdots\times H_k)\}$.