# Projective special linear group:PSL(3,5)

View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## 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