Projective special linear group of degree two
Contents
Definition
For a field or commutative unital ring
The projective special linear group of degree two over a field , or more generally over a commutative unital ring
, is defined as the quotient of the special linear group of degree two over the same field or commutative unital ring by the subgroup of scalar matrices in that group. The group is denoted by
or
.
For a prime power
Suppose is a prime power. The projective special linear group
is defined as the projective special linear group of degree two over the field (unique up to isomorphism) with
elements.
Particular cases
For prime powers 
Note that for ,
. Also, for
a power of 2 (so
),
and
.
Arithmetic functions
Below we give the arithmetic functions for , where
is a power
of a prime
.
Function | Value | Explanation |
---|---|---|
order | ![]() Becomes ![]() ![]() ![]() ![]() |
See order formulas for linear groups of degree two. Also, see element structure of projective special linear group of degree two |
number of conjugacy classes | ![]() ![]() ![]() ![]() |
See element structure of projective special linear group of degree two |
Group properties
The property listings below are for ,
a prime power.
Property | Satisfied? | Explanation |
---|---|---|
abelian group | No (never) | |
nilpotent group | No (never) | |
solvable group | No (never) | |
simple group, simple non-abelian group | Yes (almost always) | Exceptions: ![]() ![]() |
minimal simple group | Sometimes | See classification of finite minimal simple groups. Minimal simple in precisely these cases: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Elements
Over a finite field
Further information: element structure of projective special linear group of degree two over a finite field
Below is a summary:
Item | Value |
---|---|
conjugacy class sizes | Case ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Case ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Case ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
number of conjugacy classes | Case ![]() ![]() Case ![]() ![]() equals number of irreducible representations, see also linear representation theory of projective special linear group of degree two over a finite field |
number of ![]() ![]() |
![]() equals the number of irreducible representations in that characteristic, see also modular representation theory of projective special linear group of degree two over a finite field in its defining characteristic |
order | General formula: ![]() Case ![]() ![]() Case ![]() ![]() |
exponent | Case ![]() ![]() Case ![]() ![]() |
Linear representation theory
Over a finite field
Further information: linear representation theory of projective special linear group of degree two over a finite field
Below is a summary:
Item | Value |
---|---|
degrees of irreducible representations over a splitting field (such as ![]() ![]() |
Case ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Case ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Case ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
number of irreducible representations | Case ![]() ![]() ![]() ![]() See number of irreducible representations equals number of conjugacy classes, element structure of projective special linear group of degree two over a finite field#Conjugacy class structure |
quasirandom degree (minimum possible degree of nontrivial irreducible representation) | Case ![]() ![]() ![]() Case ![]() ![]() ![]() Case ![]() ![]() |
maximum degree of irreducible representation over a splitting field | ![]() |
lcm of degrees of irreducible representations over a splitting field | Case ![]() ![]() ![]() ![]() |
sum of squares of degrees of irreducible representations over a splitting field | Case ![]() ![]() ![]() ![]() equal to the group order. See sum of squares of degrees of irreducible representations equals group order |