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

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