Elementary subgroup

Builds on Ring General linear group Elementary matrix Subgroup Stable general linear group

Definition

Let $R$ be a ring with identity and let $n \ge 2$ . For $1 \le i, j \le n$ with $i \ne j$ and $r \in R$ , let $e_{ij}(r) \in GL_n(R)$ be the elementary matrix with $1$ on the diagonal, $r$ in position $(i, j)$ , and $0$ elsewhere. The elementary subgroup $E_n(R)$ is the subgroup of the general linear group $GL_n(R)$ generated by all $e_{ij}(r)$ . The stable elementary subgroup $E(R)$ is the union of the $E_n(R)$ inside the stable general linear group $GL(R)$ .

Remarks

The generators satisfy $e_{ij}(r) e_{ij}(s) = e_{ij}(r + s)$ , so $e_{ij}(r)^{-1} = e_{ij}(-r)$ . Whitehead's lemma: $E(R)$ is equal to the commutator subgroup $[GL(R), GL(R)]$ . Example: over $\Z$ , the matrix $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} = e_{12}(1)\, e_{21}(-1)\, e_{12}(1)$ lies in $E_2(\Z)$ , as multiplying out the right side shows. Non-example: $\mathrm{diag}(-1, 1) \in GL_2(\Z)$ is not in $E_2(\Z)$ , since every $e_{ij}(r)$ has determinant $1$ and this matrix has determinant $-1$ . Only the row operations "add a multiple of one row to another" are used; scaling a row and swapping two rows are not among the generators.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`WRF6FN6W1E`
URI	`m2://WRF6FN6W1E`
Kind	definition
Writing	W0 stub
MSC	19B14, 20G35
Defines	Elementary subgroup
Uses	Ring, General linear group, Elementary matrix, Subgroup, Stable general linear group
Changed	3 November 2026

Elementary subgroup

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here