projective special linear group PSL(n,q)

GPTKB entity

Predicate	Object
gptkbp:instanceOf	gptkb:group_of_people
gptkbp:actsOn	projective space of dimension n-1 over finite field of order q
gptkbp:alsoKnownAs	gptkb:PSL(n,q)
gptkbp:automorphismGroup	gptkb:projective_semilinear_group_PΓL(n,q)
gptkbp:centerOfSL(n,q)	cyclic group of order gcd(n,q-1)
gptkbp:definedIn	the quotient of the special linear group SL(n,q) by its center
gptkbp:exceptionalIsomorphisms	PSL(2,4) ≅ PSL(2,5) ≅ A_5 PSL(2,9) ≅ A_6 PSL(4,2) ≅ A_8
gptkbp:firstSimpleCase	gptkb:PSL(2,5)
gptkbp:hasSubgroup	gptkb:projective_general_linear_group_PGL(n,q)
https://www.w3.org/2000/01/rdf-schema#label	projective special linear group PSL(n,q)
gptkbp:importantFor	gptkb:algebraic_geometry combinatorics finite group theory
gptkbp:infiniteFor	infinite q
gptkbp:isFinite	finite q
gptkbp:isNonAbelian	true for n>1 and q>3
gptkbp:isomorphicTo	alternating group A_5 for PSL(2,5) alternating group A_6 for PSL(2,9)
gptkbp:isQuotientOf	gptkb:special_linear_group_SL(n,q)
gptkbp:isSimple	true, except for (n,q) = (2,2) and (2,3)
gptkbp:namedAfter	orthogonal group projective linear group
gptkbp:notation	gptkb:PSL(n,q)
gptkbp:order	q^{n(n-1)/2} \\prod_{k=2}^n (q^k-1)/d, where d=gcd(n,q-1)
gptkbp:parameter	n q
gptkbp:relatedTo	gptkb:Chevalley_groups Lie type groups
gptkbp:usedIn	classification of finite simple groups
gptkbp:bfsParent	gptkb:SL(n,q) gptkb:special_linear_group_SL(n,q)
gptkbp:bfsLayer	7