The special linear group 
over a commutative ring 
with identity is the group
of 
matrices with entries in 
and determinant 1. For a field 
,
 |
(1)
|
It is the kernel of the determinant group homomorphism from the general
linear group 
to the multiplicative group 
and is therefore a normal
subgroup.
For the finite field 
, where 
is a prime power, writing 
for the group
order of a finite group 
gives
 |
(2)
|
The group center consists of the scalar matrices 
for which 
,
where 
is the 
identity matrix. Its group
order is 
,
and the resulting quotient group is the projective
special linear group.
The group 
is a real Lie group, while
is a complex Lie group. Their dimensions
are 
over 
and 
,
respectively, and their Lie algebra consists of the
matrices having matrix trace zero,
 |
(3)
|
where 
denotes the set of all 
matrices over 
. The groups 
and 
are connected for
,
and 
is simply connected. The finite examples 
are Lie-type
groups.
, 
, 
, and 
." §2.1 in