Subgroup structure of projective special linear group of degree two over a finite field

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This article gives specific information, namely, subgroup structure, about a family of groups, namely: projective special linear group of degree two.
View subgroup structure of group families | View other specific information about projective special linear group of degree two

Particular cases

Group p q order of the group (= (q^3 - q)/2 if q odd, q^3 - q if q even) subgroup structure page
symmetric group:S3 2 2 6 subgroup structure of symmetric group:S3
alternating group:A4 3 3 12 subgroup structure of alternating group:A4
alternating group:A5 2 4 60 subgroup structure of alternating group:A5
alternating group:A5 5 5 60 subgroup structure of alternating group:A5
projective special linear group:PSL(3,2) 7 7 168 subgroup structure of projective special linear group:PSL(3,2)
projective special linear group:PSL(2,8) 2 8 504 subgroup structure of projective special linear group:PSL(2,8)
alternating group:A6 3 9 360 subgroup structure of alternating group:A6

Sylow subgroups

We consider the group PSL(2,q) over the field \mathbb{F}_q of q elements. q is a prime power of the form p^r where p is a prime number and r is a positive integer. p is hence also the characteristic of \mathbb{F}_q. We call p the characteristic prime.

Note that when p = 2 (q even), the order of the group is q^3 - q. When p \ne 2 (q odd), the order of the group is (q^3 - q)/2.

Sylow subgroups for the characteristic prime

Item Value
order of p-Sylow subgroup q = p^r
index of p-Sylow subgroup Case p = 2: q^2 - 1 = (q- 1)(q +1) = p^{2r} - 1
Case p \ne 2: (q^2 - 1)/2
explicit description of one of the p-Sylow subgroups image mod the center of unitriangular matrix group of degree two: \{ \begin{pmatrix} 1 & b \\ 0 & 1 \\\end{pmatrix} \mid b \in \mathbb{F}_q \}
Note that this subgroup intersects the center trivially, so there are no identifications when taking the quotient by the center
isomorphism class of p-Sylow subgroup additive group of \mathbb{F}_q, which is an elementary abelian group of order q = p^r, i.e., a direct product of r copies of the cyclic group of order p
explicit description of p-Sylow normalizer image mod the center of Borel subgroup of degree two: \{ \begin{pmatrix} a & b \\ 0 & a^{-1} \\\end{pmatrix} \mid a \in \mathbb{F}_q^\ast, b \in \mathbb{F}_q \}. Explicitly, we identify the matrix up to sign on the whole matrix.
isomorphism class of p-Sylow normalizer It is the external semidirect product of \mathbb{F}_q by the quotient \mathbb{F}_q^\ast/\{ \pm 1 \} where the latter acts on the former via the multiplication action of the square of the acting element. Note that the action is well defined because the square of \pm 1 acts trivially.
Equivalently, it is the obvious subgroup \mathbb{F}_q \rtimes (\mathbb{F}_q^\ast)^2 of the general affine group of degree one GA(1,q).
For p = 2 (so q = 2,4,8,\dots), it is isomorphic to the general affine group of degree one GA(1,q).
For q = 3, it is cyclic group:Z3 and for q = 5, it is dihedral group:D10.
order of p-Sylow normalizer Case p = 2, i.e., q even: q(q - 1)
Case p \ne 2, i.e., q odd: q(q  - 1)/2
p-Sylow number (i.e., number of p-Sylow subgroups) = index of p-Sylow normalizer q + 1 (congruent to 1 mod p, as expected from the congruence condition on Sylow numbers)

Sylow subgroups for other primes: cases and summary

For any prime \ell, the \ell-Sylow subgroup is nontrivial iff \ell \mid q^3 - q. If \ell \ne p, then it does not divide q, so we get that \ell \mid q^2 - 1 which means that either \ell \mid q - 1 or \ell \mid q + 1. Further, if \ell \ne 2, exactly one of these cases can occur. For \ell = 2, we make cases based on the residue of q mod 8. The summary of cases is below and more details are in later sections.

Case on \ell and q Isomorphism type of \ell-Sylow subgroup Isomorphism type of \ell-Sylow normalizer Order of \ell-Sylow normalizer \ell-Sylow number = index of \ell-Sylow normalizer
\ell is an odd prime dividing q - 1, p = 2 cyclic group dihedral group 2(q - 1) q(q + 1)/2
\ell is an odd prime dividing q - 1, p \ne 2 cyclic group dihedral group q - 1 q(q + 1)/2
\ell is an odd prime dividing q + 1, p = 2 cyclic group dihedral group 2(q + 1) q(q - 1)/2
\ell is an odd prime dividing q + 1, p \ne 2 cyclic group dicyclic group q + 1 q(q - 1)/2
\ell = 2 and q \equiv 1 \pmod 8 dihedral group dihedral group largest power of 2 dividing the order = twice the largest power of 2 dividing q - 1 largest odd number dividing the order
\ell = 2 and q \equiv 5 \pmod 8 Klein four-group alternating group:A4 12 (q^3 - q)/24
\ell = 2 and q \equiv 7 \pmod 8 dihedral group dihedral group largest power of 2 dividing the order = twice the largest power of 2 dividing q + 1 largest odd number dividing the order
\ell = 2 and q \equiv 3 \pmod 8 Klein four-group alternating group:A4 12 (q^3 - q)/24