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] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
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] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
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

Arithmetic functions of a counting nature

Function	Value	Similar groups	Explanation for function value
number of conjugacy classes	13	groups with same order and number of conjugacy classes | groups with same number of conjugacy classes	See element structure of projective special linear group of degree two over a finite discrete valuation ring and element structure of projective special linear group:PSL(2,Z9)
number of conjugacy classes of subgroups	32	groups with same order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups	See subgroup structure of projective special linear group:PSL(2,Z9)
number of subgroups	340	groups with same order and number of subgroups | groups with same number of subgroups	See subgroup structure of projective special linear group:PSL(2,Z9)

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.