McLain's group

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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

Online references