Subnormal subgroups of En(R) have no free, non-abelian quotients, when n≧3

Abstract

It is known that for certain rings R (for example R = ℤ, the ring of rational integers) the group GL2(R) contains subnormal subgroups which have free, non-abelian quotients. When such a subgroup has finite index it follows that every countable group is embeddable in a quotient of GL2(R). (In this case GL2(R) is said to be SQ-universal.) In this note we prove that the existence of subnormal subgroups of GL2(R) with this property is a phenomenon peculiar to “n = 2”.For a large class of rings (which includes all commutative rings) it is shown that, for all , no subnormal subgroup of En(R) has a free, non-abelian quotient, when n≧3. (En(R) is the subgroup of GLn(R) generated by the elementary matrices.) In addition it is proved that, if is an SRt-ring, for some t ≧ 2, then no subnormal subgroup of GLn(R) has a free, non-abelian quotient, when n ≧ max (t, 3). From the above these results are best possible since ℤ is an SR3 ring.