On free subgroups of units in quaternion algebras

Jan Krempa. "On free subgroups of units in quaternion algebras." Colloquium Mathematicae 88.1 (2001): 21-27. <http://eudml.org/doc/284229>.

@article{JanKrempa2001,
abstract = {It is well known that for the ring H(ℤ) of integral quaternions the unit group U(H(ℤ) is finite. On the other hand, for the rational quaternion algebra H(ℚ), its unit group is infinite and even contains a nontrivial free subgroup. In this note (see Theorem 1.5 and Corollary 2.6) we find all intermediate rings ℤ ⊂ A ⊆ ℚ such that the group of units U(H(A)) of quaternions over A contains a nontrivial free subgroup. In each case we indicate such a subgroup explicitly. We do our best to keep the arguments as simple as possible.},
author = {Jan Krempa},
journal = {Colloquium Mathematicae},
keywords = {Cayley algebras; groups of units; free groups; Tits alternative; quaternion rings},
language = {eng},
number = {1},
pages = {21-27},
title = {On free subgroups of units in quaternion algebras},
url = {http://eudml.org/doc/284229},
volume = {88},
year = {2001},
}

TY - JOUR
AU - Jan Krempa
TI - On free subgroups of units in quaternion algebras
JO - Colloquium Mathematicae
PY - 2001
VL - 88
IS - 1
SP - 21
EP - 27
AB - It is well known that for the ring H(ℤ) of integral quaternions the unit group U(H(ℤ) is finite. On the other hand, for the rational quaternion algebra H(ℚ), its unit group is infinite and even contains a nontrivial free subgroup. In this note (see Theorem 1.5 and Corollary 2.6) we find all intermediate rings ℤ ⊂ A ⊆ ℚ such that the group of units U(H(A)) of quaternions over A contains a nontrivial free subgroup. In each case we indicate such a subgroup explicitly. We do our best to keep the arguments as simple as possible.
LA - eng
KW - Cayley algebras; groups of units; free groups; Tits alternative; quaternion rings
UR - http://eudml.org/doc/284229
ER -