Projective special linear group:PSL(3,5)

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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:F5, denoted PSL(3,5).
  2. It is the special linear group of degree three over field:F5, denoted SL(3,5).
  3. It is the projective general linear group of degree three over field:F5, denoted PGL(3,5).

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) 372000 groups with same order As SL(3,q), q = 5: q^3(q^3 - 1)(q^2 - 1) = 5^3(5^3 - 1)(5^2 - 1) = (125)(124)(24) = 372000
exponent of a group 3720 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 conjugacy classes 30 As SL(3,q), q = 5: q^2 + q = 5^2 + 5 = 30

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 No See classification of finite minimal simple groups

GAP implementation

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