Subgroup structure of projective general 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 general linear group of degree two.
View subgroup structure of group families | View other specific information about projective general linear group of degree two

Particular cases

Group p q Order of the group Subgroup structure page
symmetric group:S3 2 2 6 subgroup structure of symmetric group:S3
symmetric group:S4 3 3 24 subgroup structure of symmetric group:S4
alternating group:A5 2 4 60 subgroup structure of alternating group:A5
symmetric group:S5 5 5 120 subgroup structure of symmetric group:S5
projective general linear group:PGL(2,7) 7 7 336 subgroup structure of projective general linear group:PGL(2,7)
projective general linear group:PGL(2,9) 3 9 720 subgroup structure of projective general linear group:PGL(2,9)

Sylow subgroups

We consider the group PGL(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.

Sylow subgroups for the characteristic prime

Item Value
order of p-Sylow subgroup q = p^r
index of p-Sylow subgroup q^2 - 1 = (q- 1)(q +1) = p^{2r} - 1
explicit description of one of the p-Sylow subgroups image of unitriangular matrix group of degree two: \{ \begin{pmatrix} 1 & b \\ 0 & 1 \\\end{pmatrix} \mid b \in \mathbb{F}_q \}
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 of Borel subgroup of degree two in GL(2,q): \{ \begin{pmatrix} a & b \\ 0 & c \\\end{pmatrix} \mid a,c \in \mathbb{F}_q^\ast, b \in \mathbb{F}_q \}
isomorphism class of p-Sylow normalizer general affine group of degree one GA(1,q)
order of p-Sylow normalizer q(q - 1) = p^{2r} - p^r
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 4:

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 2(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 dihedral group 2(q + 1) q(q - 1)/2
\ell = 2 and q \equiv 1 \pmod 4 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 4 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