Special linear group:SL(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

The group, denoted $SL(2,\mathbb {Z} _{9})$ or $SL(2,\mathbb {Z} /9\mathbb {Z} )$, is defined as the 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 648#Arithmetic functions

Basic arithmetic functions

Function	Value	Similar groups	Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set)	648	groups with same order	As $SL(2,\mathbb {Z} _{9})$, where $\mathbb {Z} _{9}$ is a DVR of length $l=2$ over a field of size $q=3$: $q^{3l-2}(q^{2}-1)=3^{3(2)-2}(3^{2}-1)=(81)(8)=648$

Arithmetic functions of a counting nature

Function	Value	Similar groups	Explanation for function value
number of conjugacy classes	25	groups with same order and number of conjugacy classes | groups with same number of conjugacy classes	As $SL(2,R)$, $R$ a DVR of length $l=2$ over a field of size $q$, $q$ odd: $q^{l}+4{\frac {q^{l}-1}{q-1}}=3^{2}+4(8)/2=9+16=25$ (note that for $l=2$, the formula simplifies to $(q+2)^{2}$) See element structure of special linear group of degree two over a finite discrete valuation ring, element structure of special linear group:SL(2,Z9)
number of conjugacy classes of subgroups	48	groups with same order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups	See subgroup structure of special linear group:SL(2,Z9)
number of subgroups	456	groups with same order and number of subgroups | groups with same number of subgroups	See subgroup structure of special linear group:SL(2,Z9)

GAP implementation

Group ID

This finite group has order 648 and has ID 641 among the groups of order 648 in GAP's SmallGroup library. For context, there are groups of order 648. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(648,641)

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

gap> G := SmallGroup(648,641);

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

IdGroup(G) = [648,641]

or just do:

IdGroup(G)

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

Other descriptions

Description	Functions used
SL(2,ZmodnZ(9))	SL, ZmodnZ