# Projective special linear group:PSL(2,19)

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## Definition

This finite group is defined as the projective special linear group of degree two over field:F19, the field with 19 elements.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 3420#Arithmetic functions

### Basic arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 3420 groups with same order As $PSL(2,q)$, $q = 19$: $(q^3 - q)/2 = q(q-1)(q+1)/2 = (19^3 - 19)/2 = 19(19-1)(19+1)/2 = 3420$
exponent of a group 1710 groups with same order and exponent of a group | groups with same exponent of a group As $PSL(2,q)$, $q = 19$, $p = 19$ where $p$ is the characteristic: $p(q^2 - 1)/4 = 19(19^2 - 1)/4 = 1710$

### Arithmetic functions of a counting nature

Function Value Similar groups Explanation
number of conjugacy classes 12 groups with same order and number of conjugacy classes | groups with same number of conjugacy classes As $PSL(2,q), q = 19$ ( $q$ odd): $(q + 5)/2 = (19 + 5)/2 = 12$
See element structure of projective special linear group of degree two over a finite field, element structure of projective special linear group:PSL(2,19)

## Group properties

Property Satisfied? Explanation Corollary properties satisfied/dissatisfied
simple group, simple non-abelian group Yes projective special linear group is simple except in finitely many cases, but this isn't one of the finite exceptions
minimal simple group No Contains subgroup isomorphic to alternating group:A5. See also classification of finite minimal simple groups
solvable group No Dissatisfies: nilpotent group, abelian group

## GAP implementation

Description Functions used
PSL(2,19) PSL