General linear group:GL(2,Z9)

View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

Definition

This group can be defined in the following equivalent ways:

1. It is the group $GL(2,\mathbb{Z}_9)$ or $GL(2,\mathbb{Z}/9\mathbb{Z})$, i.e., the general linear group of degree two over the ring of integers modulo $9$.
2. It is the group $GL(2,\mathbb{F}_3[t]/(t^2))$, i.e., the general linear group of degree two over the ring $\mathbb{F}_3[t]/(t^2)$.

Note that although the rings in question are different, the corresponding general linear groups are isomorphic.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 3888#Arithmetic functions

Basic arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 3888 groups with same order As $GL(2,R)$, $R$ a discrete valuation ring of length $l = 2$ with residue field of size $q = 3$: $q^{4l - 3}(q - 1)(q^2 - 1) = 3^5(2)(8) = 3888$
exponent of a group 72 groups with same order and exponent of a group | groups with same exponent of a group
nilpotency class -- not a nilpotent group
derived length 4 groups with same order and derived length | groups with same derived length

GAP implementation

Other descriptions

Description Functions used
GL(2,ZmodnZ(9)) GL and ZmodnZ