Projective special linear group:PSL(2,19)

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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 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