Rigid metabelian pro-$p$-groups

S. G. Afanas'eva ; N. S. Romanovskii

Algebra i logika, Tome 53 (2014) no. 2, pp. 162-177

Résumé

A metabelian pro-$p$-group $G$ is rigid if it has a normal series of the form $$ G=G_1\ge G_2\ge G_3=1 $$ such that the factor group $A=G/G_2$ is torsion-free Abelian and $C=G_2$ is torsion-free as a $\mathbb Z_pA$-module. If $G$ is a non-Abelian group, then the subgroup $G_2$, as well as the given series, is uniquely defined by the properties mentioned. An Abelian pro-$p$-group is rigid if it is torsion-free, and as $G_2$ we can take either the trivial subgroup or the entire group. We prove that all rigid $2$-step solvable pro-$p$-groups are mutually universally equivalent. Rigid metabelian pro-$p$-groups can be treated as $2$-graded groups with possible gradings $(1,1)$, $(1,0)$, and $(0,1)$. If a group is $2$-step solvable, then its grading is $(1,1)$. For an Abelian group, there are two options: namely, grading $(1,0)$, if $G_2=1$, and grading $(0,1)$ if $G_2=G$. A morphism between $2$-graded rigid pro-$p$-groups is a homomorphism $\varphi\colon G\to H$ such that $G_i\varphi\le H_i$. It is shown that in the category of $2$-graded rigid pro-$p$-groups, a coproduct operation exists, and we establish its properties.

Détail

Keywords: rigid metabelian pro-$p$-group, $2$-graded group.