# Projective special linear group:PSL(2,Z9)

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## Definition

This group is defined as the projective special linear group of degree two over the ring of integers modulo 9.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 324#Arithmetic functions

### Basic arithmetic functions

Function Value Similar groups Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set) 324 groups with same order As $PSL(2,R)$, $R$ a DVR of length $l = 2$ over a field of size $q = 3$, $R$ has $m = 2$ square roots of unity (because field characteristic is odd):
$q^{3l - 2}(q^2 - 1)/m = 3^{3(2) - 2}(3^2 - 1)/2 = 324$
exponent of a group 18 groups with same order and exponent of a group | groups with same exponent of a group PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
nilpotency class -- -- not a nilpotent group
derived length 3 groups with same order and derived length | groups with same derived length PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Frattini length 1 groups with same order and Frattini length | groups with same Frattini length Frattini-free group
Fitting length 3 groups with same order and Fitting length | groups with same Fitting length

## GAP implementation

### Group ID

This finite group has order 324 and has ID 160 among the groups of order 324 in GAP's SmallGroup library. For context, there are groups of order 324. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(324,160)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(324,160);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [324,160]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.