# McLain's group

From Groupprops

## Definition

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

as the group of automorphisms of generated by the elementary matrices:

McLain's group over a ring is defined as the group , where is the set of rational numbers with its usual total ordering.

McLain's group is denoted or sometimes simply . It is typically considered in cases where is a field or division ring. Both the characteristic zero case (e.g., ) and the characteristic case (e.g., ) 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 0387944613^{More info} |
348 | Section 12.1 | formal definition introduced as 12.1.9 | Google Books |

### Online references

- MathOverflow question: A question about matrices: The questions asks for details on McLain's group. Person asking the question is unable to access the original McLain paper with the details