Projective general linear group of degree two
Contents
Definition
For a field , the projective general linear group of degree two
or
is defined as the quotient group of the general linear group of degree two
by its center, which is the group of scalar matrices in it (because center of general linear group is group of scalar matrices over center).
In other words, it is the inner automorphism group of the general linear group of degree two.
The projective general linear group of degree two is the degree two special case of the projective general linear group.
For a prime power, the projective general linear group of degree two denoted
is defined as the projective general linear group of degree two over the field
of
elements (unique up to field isomorphism).
Arithmetic functions
Over a finite field
Here, is the size of the finite field for which we consider the group
.
is the characteristic of the field, so
is a power of
.
Function | Value | Explanation |
---|---|---|
order | ![]() |
![]() ![]() ![]() |
exponent | ![]() ![]() ![]() ![]() |
Elements of order ![]() ![]() ![]() |
number of conjugacy classes | ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
Particular cases
Finite fields
Note that for ,
is isomorphic to all of
. For
a power of 2,
is isomorphic to
and
but not to
.
For not a power of
,
admits
as a subgroup of index two.
and
have the same order, and in fact the same composition factors, but are non-isomorphic, because
admits
as a quotient rather than as a subgroup. In general, for
,
is an almost simple group and
is a quasisimple group.
Field size ![]() |
Field characteristic ![]() |
Exponent on ![]() ![]() |
Projective general linear group ![]() |
Order (= ![]() |
Second part of GAP ID | Proof of isomorphism | Comments |
---|---|---|---|---|---|---|---|
2 | 2 | 1 | symmetric group:S3 | 6 | 1 | PGL(2,2) is isomorphic to S3 | not an almost simple group (one of two exceptions) |
3 | 3 | 1 | symmetric group:S4 | 24 | 12 | PGL(2,3) is isomorphic to S4 | not an almost simple group (one of two exceptions) |
4 | 2 | 2 | alternating group:A5 | 60 | 5 | PGL(2,4) is isomorphic to A5 | a simple non-abelian group (as is the case for all powers of 2 with exponent at least 2) |
5 | 5 | 1 | symmetric group:S5 | 120 | 34 | PGL(2,5) is isomorphic to S5 | an almost simple group for the simple non-abelian group alternating group:A5. |
7 | 7 | 1 | projective general linear group:PGL(2,7) | 336 | 208 | -- | an almost simple group for the simple non-abelian group PSL(2,7) (which is isomorphic to PSL(3,2)). |
8 | 2 | 3 | projective special linear group:PSL(2,8) | 504 | 156 | see note above on ![]() ![]() |
a simple non-abelian group (as is the case for all powers of 2 with exponent at least 2) |
9 | 3 | 2 | projective general linear group:PGL(2,9) (note: this is not isomorphic to symmetric group:S6) | 720 | 764 | -- | an almost simple group for the simple non-abelian group PSL(2,9), which is isomorphic to alternating group:A6. There are two other almost simple groups of the same order for ![]() ![]() |
11 | 11 | 1 | projective general linear group:PGL(2,11) | 1320 | 133 | -- | an almost simple group for projective special linear group:PSL(2,11). |
Elements
Over a finite field
Further information: element structure of projective general linear group of degree two over a finite field
Linear representation theory
Over a finite field
Further information: linear representation theory of projective general linear group of degree two over a finite field
Below is a summary of the linear representation theory of :
Item | Value |
---|---|
degrees of irreducible representations over a splitting field | Case ![]() ![]() ![]() ![]() ![]() ![]() Case ![]() ![]() ![]() ![]() ![]() ![]() |
number of irreducible representations | Case ![]() ![]() ![]() ![]() See number of irreducible representations equals number of conjugacy classes, element structure of projective general linear group of degree two over a finite field#Conjugacy class structure |
quasirandom degree (minimum degree of nontrivial ireducible representation) | 1 |
maximum degree of irreducible representation | ![]() |
lcm of degrees of irreducible representations | Case ![]() ![]() ![]() ![]() |
sum of squares of degrees of irreducible representations | ![]() |
Subgroups
Further information: subgroup structure of projective general linear group of degree two over a finite field