Projective special linear group:PSL(3,3)

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
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 is a finite group defined in the following equivalent ways:

1. It is the projective special linear group of degree three over field:F3, denoted $PSL(3,3)$.
2. It is the special linear group of degree three over field:F3, denoted $SL(3,3)$.
3. It is the projective general linear group of degree three over field:F3, denoted $PGL(3,3)$.

Equivalence of definitions

The equivalence of definitions follows from isomorphism between linear groups when degree power map is bijective.

Arithmetic functions

Basic arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	5616	groups with same order	As $SL(3,q),q=3$: $q^{3}(q^{3}-1)(q^{2}-1)=3^{3}(3^{3}-1)(3^{2}-1)=(27)(26)(8)=5616$
exponent of a group	312	groups with same order and exponent of a group | groups with same exponent of a group
Frattini length	1	groups with same order and Frattini length | groups with same Frattini length	As the group is a simple non-abelian group, its Frattini length must be one.

Arithmetic functions of a counting nature

Function	Value	Explanation
number of subgroups	6374
number of conjugacy classes	12	As $SL(3,q),q=3$: $q^{2}+q=3^{2}+3=12$
number of conjugacy classes of subgroups	51

Group properties

Property	Satisfied?	Explanation
abelian group	No
nilpotent group	No
solvable group	No
simple group, simple non-abelian group	Yes	projective special linear group is simple
minimal simple group	Yes

GAP implementation

Description	Functions used
PSL(3,3)	PSL
SL(3,3)	SL
PGL(3,3)	PGL