# Special linear group

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of degree (order) over a ring The subgroup of the general linear group which is the kernel of a determinant homomorphism . The structure of depends on , and the type of determinant defined on . There are three main types of determinants of importance here: the ordinary determinant in the case when is a commutative ring, the non-commutative Dieudonné determinant (cf. Determinant) when is a division ring (see ), and the reduced norm homomorphism for a division ring that is finite dimensional over its centre (see ). has the following noteworthy subgroups: the group generated by the elementary matrices (see Algebraic -theory) and, for every two-sided ideal of , the congruence subgroup and the group which is the normal subgroup of generated by the matrices for . Let and let be the imbedding of into . Then passage to the direct limit gives the group . The group is defined in a similar way. When one writes and instead of and , respectively. The latter is called the stable special linear group of the ring . The normal subgroup structure of is closely connected with the structure of the groups : A subgroup is normal in if and only if, for some (unique) two-sided ideal of , the following inclusions hold: Thus, the Abelian groups classify the normal subgroups of . The group is called the reduced Whitehead group of . A satisfactory description of the normal subgroup structure of for an arbitrary ring uses a condition involving the stable rank of the ideal ( ). Namely, if , then there is an isomorphism In addition, if the conditions , hold, then for every normal subgroup of the inclusions hold for an appropriate , where , and is the pre-image of the centre of in . For certain special rings definitive results are known (see , , for example).

In the case of the non-commutative Dieudonné determinant (so that is a division ring), the results are exhaustive. The groups and coincide. is the commutator subgroup of , except in the case of (where denotes the field of elements). The centre of consists of the scalar matrices , where is an element of the centre of and , being the commutator subgroup of the multiplicative group of the division ring . The quotient group is simple except when and . When , and is isomorphic to the symmetric group of degree 3, while is isomorphic to the alternating group of degree 4.

If is a reduced norm homomorphism, then and so that the group is trivial when is a field. The conjecture that for any division ring stood for a long time. However, in 1975 it was shown that this is not true (see ). The groups play an important role in algebraic geometry (see , ). There are also generalizations of the reduced norm homomorphism, which have stimulated a series of new investigations into special linear groups.

How to Cite This Entry:
Special linear group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Special_linear_group&oldid=18074
This article was adapted from an original article by V.I. Yanchevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article