Unitriangular matrix group
Definition as matrix group
Suppose is a commutative unital ring and is a natural number. The unitriangular matrix group, denoted , , or , is the group, under multiplication, with s on the diagonal, s below the diagonal, and arbitrary entries above the diagonal.
Note that the symbol is also used for the unitary group, hence we use or to avoid confusion.
Definition by presentation
The presentation given here is similar to the presentation used for the Steinberg group over a unital ring. Specifically, we use only those generators and relations that correspond to upper triangular matrices and discard the rest. We denote by the matrix with 1s on the diagonal, in the entry, and zeros elsewhere. Then:
- The following is a generating set: For every element and for , include the element .
- We can get a presentation of the group using this generating set, by including the following relations:
- (note that this in particular implies that is the identity element for all ).
- (the Steinberg group version also imposes the condition , but this constraint follows automatically in the upper triangular case because and ).
- (i.e., is the identity element) for .
Note that this presentation can be trimmed quite a bit. In fact, if is a generating set for the additive group of , the set:
is a generating set for , and we can work out a presentation in terms of this generating set using the relations above.
Case of a field
When is a field, the unitriangular matrix group can also be described as a maximal unipotent subgroup of the general linear group . It is also a maximal unipotent subgroup of the special linear group .
Case of a finite field
When is a finite field with elements and characteristic (so is a power of ), then is also denoted , and is a -Sylow subgroup of .
|Value of||(the "dimension" of , the size is )||Nature of unitriangular matrix group over a ring|
|2||1||isomorphic to the additive group of|
|3||3||see unitriangular matrix group of degree three|
|4||6||see unitriangular matrix group of degree four|
We give here the arithmetic functions for .
|derived length||(the smallest integer greater than or equal to|
|Frattini length||PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] (depends both on the nature of and on )|