Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

Contents 1 Definition 2 Arithmetic functions 3 Group properties 4 Elements

Definition

The special linear group of degree two over a field k, or more generally over a commutative unital ring R, is defined as the group of matrices with determinant 1 under matrix multiplication, and entries over R . The group is denoted by SL(2,R) or SL₂(R).

When q is a prime power, SL(2,q) is the special linear group of degree two over the field (unique up to isomorphism) with q elements.

The underlying set of the group is:

.

The group operation is given by:

.

The identity element is:

.

The inverse map is given by:

Arithmetic functions

Here, q denotes the order of the finite field and the group we work with is GL(2,q). p is the characteristic of the field, i.e., it is the prime whose power q is.

Function	Value	Explanation
order		Kernel of determinant map from group of size q(q + 1)(q − 1)2 surjecting to group of size q − 1.
exponent	p(q2 − 1) if p = 2, if p > 2	There are elements of order p,q − 1,q + 1, orders of all elements divide one of these.
number of conjugacy classes	q + 1 if p = 2, q + 4 if p > 2	For p > 2, q semisimple conjugacy classes (that do not split from GL(2,q) and four conjugacy classes that merge into two in GL(2,q).

Group properties

Property	Satisfied	Explanation
Abelian group	Yes if q = 2, no otherwise
Nilpotent group	Yes if q = 2, no otherwise	special linear group is perfect for , the case of q = 2,3 can be checked.
Solvable group	Yes if q = 2,3, no otherwise.	special linear group is perfect for , the case of q = 2,3 can be checked.
Supersolvable group	Yes if q = 2, no otherwise	special linear group is perfect for , the case of q = 2,3 can be checked.
Quasisimple group	Yes if	special linear group is quasisimple for .

Elements

Further information: Element structure of special linear group of degree two