Constructible Matrix Groups

V. A. Roman'kov ; N. G. Khisamiev

Algebra i logika, Tome 43 (2004) no. 5, pp. 603-613

Résumé

We prove that the additive group of a ring $K$ is constructible if the group $GL_2(K)$ is constructible. It is stated that under one extra condition on $K$, the constructibility of $GL_2(K)$ implies that $K$ is constructible as a module over its subring $L$ generated by all invertible elements of the ring $K$; this is true, in particular, if $K$ coincides with $L$, for instance, if $K$ is a field or a group ring of an Abelian group with the specified property. We construct an example of a commutative associative ring $K$ with 1 such that its multiplicative group $K^{\ast}$ is constructible but its additive group is not. It is shown that for a constructible group $G$ represented by matrices over a field, the factors w. r. t. members of the upper central series are also constructible. It is proved that a free product of constructible groups is again constructible, and conditions are specified under which relevant statements hold of free products with amalgamated subgroup; this is true, in particular, for the case where an amalgamated subgroup is finite. Also we give an example of a constructible group $GL_2(K)$ with a non-constructible ring $K$. Similar results are valid for the case where the group $SL_2(K)$ is treated in place of $GL_2(K)$.

Détail

Mots-clés : constructible matrix group
Keywords: commutative associative ring.