# McLain's group

## Definition

McLain's group is a particular kind of unitriangular matrix group of infinite degree over a ring. Specifically, for a ring $R$ and a partially ordered indexing set $I$, define:

$UT(I,R)$

as the group of automorphisms of $R^I$ generated by the elementary matrices:

$e_i \mapsto e_i + re_j, \qquad i,j \in I, i < j, r \in R$

McLain's group over a ring $R$ is defined as the group $UT(\mathbb{Q}, R)$, where $\mathbb{Q}$ is the set of rational numbers with its usual total ordering.

McLain's group is denoted $M(\mathbb{Q}, R)$ or sometimes simply $M$. It is typically considered in cases where $R$ is a field or division ring. Both the characteristic zero case (e.g., $M(\mathbb{Q}, \mathbb{Q})$) and the characteristic $p$ case (e.g., $M(\mathbb{Q}, \mathbb{F}_p)$) generate groups of interest.

## References

### Textbook references

Book Page number Chapter and section Contextual information View
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613More info 348 Section 12.1 formal definition introduced as 12.1.9 Google Books