General linear group:GL(2,Z9)

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

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