# Projective special linear group:PSL(2,27)

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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 27 elements.

## Arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 9828 groups with same order As $PSL(2,q)$, $q = 27$ ( $q$ odd): $(q^3 - q)/2 = q(q-1)(q+1)/2 = (27^3 - 27)/2 = 27(27-1)(27+1)/2 = 9828$
exponent of a group 546 groups with same order and exponent of a group | groups with same exponent of a group As $PSL(2,q)$, $q = 27$, $p = 3$ where $p$ is the characteristic: $p(q^2 - 1)/4 = 3(27^2 - 1)/4 = 546$

## 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