General linear group:GL(2,Z)
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Definition
The group is defined as the group of invertible
matrices over the ring of integers, under matrix multiplication. Since the determinant is multiplicative and the only invertible integers are
, this can equivalently be defined as:
.
This is a particular case of a general linear group over integers, which in turn is a particular case of a general linear group.
The subgroup of matrices of determinant one is special linear group:SL(2,Z), and it is a subgroup of index two.
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order | Infinite (countable) | |
exponent | Infinite (countable) | |
derived length | not defined | |
Frattini length | not defined | Has a free non-abelian subgroup, so not solvable. |
Group properties
Property | Satisfied | Explanation | Comment |
---|---|---|---|
abelian group | No | ||
nilpotent group | No | ||
solvable group | No | ||
perfect group | No |
GAP implementation
The group can be defined using GAP's GeneralLinearGroup function, as:
GL(2,Integers)