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

Published online by Cambridge University Press:  20 January 2009

A. W. Mason
A. W. Mason
Affiliation:
Department of MathematicsUniversity Of Glasgow University GardensGlasgow G12 8QW
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Abstract

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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.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 34 , Issue 1 , February 1991 , pp. 113 - 119
Copyright
Copyright © Edinburgh Mathematical Society 1991

References

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