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

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $GL_2(K)$.
Mots-clés : constructible matrix group
Keywords: commutative associative ring. 
@article{AL_2004_43_5_a5,
     author = {V. A. Roman'kov and N. G. Khisamiev},
     title = {Constructible {Matrix} {Groups}},
     journal = {Algebra i logika},
     pages = {603--613},
     publisher = {mathdoc},
     volume = {43},
     number = {5},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2004_43_5_a5/}
}
TY  - JOUR
AU  - V. A. Roman'kov
AU  - N. G. Khisamiev
TI  - Constructible Matrix Groups
JO  - Algebra i logika
PY  - 2004
SP  - 603
EP  - 613
VL  - 43
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2004_43_5_a5/
LA  - ru
ID  - AL_2004_43_5_a5
ER  - 
%0 Journal Article
%A V. A. Roman'kov
%A N. G. Khisamiev
%T Constructible Matrix Groups
%J Algebra i logika
%D 2004
%P 603-613
%V 43
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2004_43_5_a5/
%G ru
%F AL_2004_43_5_a5
V. A. Roman'kov; N. G. Khisamiev. Constructible Matrix Groups. Algebra i logika, Tome 43 (2004) no. 5, pp. 603-613. http://geodesic.mathdoc.fr/item/AL_2004_43_5_a5/

[1] V. A. Romankov, N. G. Khisamiev, “O konstruktivnykh matrichnykh i uporyadochivaemykh gruppakh”, Algebra i logika, 43:3 (2004), 261–290 | MR

[2] V. Roman'kov, N. Khisamiev, “On constructible matrix or ordered groups”, Proc. Workshop “Computability and Models” (Almaty, June 24–28, 2002), ed. S. A. Badaeva, Kazak universiteti, Almaty, 2002, 44–47

[3] A. I. Maltsev, “Ob elementarnykh svoistvakh lineinykh grupp”, Nekotorye problemy matematiki i mekhaniki, izd-vo SO AN SSSR, Novosibirsk, 1961, 110–132

[4] R. Lindon, P. Shupp, Kombinatornaya teoriya grupp, Mir, M., 1980 | MR