Introduction
Let R be a commutative ring with identity, n> 3. Suslin has shown that the elementary subgroup E(n, R) is normal in the general linear group GL(n, R). In other words, E(n, R) is invariant under change of co-ordinates. Here we will establish the analogue for the Steinberg group St(n, R), when n>4. We will give a presentation for St(n, R) which is invariant under change of co-ordinates. Thus a change of co-ordinates, given by an element M of GL(n, R), will induce an automorphism aye of St(n, R). This OIM is compatible with inner conjugation by M in GL(n, R). If M is the image of some element x of Xt(n, R) then OIM is just inner conjugation by x. It follows that &(n, R) is central in St(n, R), and, if n>5, that St(n, R) is the universal central extension of E(n, R).
I am indebted to Keith Dennis for suggesting this work and formulating relevant questions when it was in progress. 9 2. THE RESULTS
2.1. Throughout
R is a commutative ring with identity. (For noncommutative rings the proofs fail, especially in 3.2). Let n>4.
DEFINITIONS.
Let U be the set of pairs (i, j) with i a unimodular column of length n, j a row of length n such that ji= 0. For (i, j) E U we put e(i, j) = 1 +ij, where 1 is the identity matrix in GL(n, R). So e(i, j)v=v+i(jv), if v is a column of length n. (Note that jv E R). And also w e(i, j)= w+ (&)j, if w is a row of length n. We have (ij)a= 0, so e(i, j) E GL(n, R).
2.2. DEFIXITION.
St*(n, R) is the group defined by the following presentation.
Generators: X(i, j) with (i, j) E U. Relations :
X(i, j)X(i, k) =X(i, j+k) if (i, j), (i, Zc) E U. X(i, j)X(k, Z)X(i, j)-l=X(k+i(jk), Z-(Zi)j), if (i, j), (k, I) E U.
Note that X(k+i(jk), I-(Zi)j)=X(e(i, j)k, Z e(i, j)-1).