Projective special linear group:PSL(2,25)

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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 group is defined as the projective special linear group of degree two over field:F25.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 7800#Arithmetic functions

Basic arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	7800	groups with same order	As $PSL(2,q),q=25$ ($q$ odd): $(q^3-q)/2=q(q-1)(q+1)/2=(25)(24)(26)/2=7800$
exponent of a group	780	groups with same order and exponent of a group | groups with same exponent of a group	As $PSL(2,q)$, $q=25$, $p=5$ where $p$ is the characteristic: $p(q^2-1)/4=5(25^2-1)/4=780$

Arithmetic functions of a counting nature

Function	Value	Similar groups	Explanation
number of conjugacy classes	15	groups with same order and number of conjugacy classes | groups with same number of conjugacy classes	As $PSL(2,q),q=25$ ($q$ odd): $(q+5)/2=(25+5)/2=15$ See element structure of projective special linear group of degree two over a finite field, element structure of projective special linear group:PSL(2,25)

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 alternating group:A5 as $PSL(2,5)$ inside it. See classification of finite minimal simple groups
solvable group	No		Dissatisfies: nilpotent group, abelian group

GAP implementation

Short descriptions

Description	Functions used
PSL(2,25)	PSL
PerfectGroup(7800,1)	PerfectGroup