Special Linear Group - Generators and Relations

Generators and Relations

If working over a ring where SL is generated by transvections (such as a field or Euclidean domain), one can give a presentation of SL using transvections with some relations. Transvections satisfy the Steinberg relations, but these are not sufficient: the resulting group is the Steinberg group, which is not the special linear group, but rather the universal central extension of the commutator subgroup of GL.

A sufficient set of relations for SL(n, Z) for n ≥ 3 is given by two of the Steinberg relations, plus a third relation (Conder, Robertson & Williams 1992, p. 19). Let Tij := eij(1) be the elementary matrix with 1's on the diagonal and in the ij position, and 0's elsewhere (and i ≠ j). Then

\begin{align} \left &= T_{ik} && \mbox{for } i \neq k\\ \left &= \mathbf{1} && \mbox{for } i \neq l, j \neq k\\ (T_{12}T_{21}^{-1}T_{12})^4 &= \mathbf{1}\\ \end{align}

are a complete set of relations for SL(n, Z), n ≥ 3.

Read more about this topic:  Special Linear Group

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