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

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

## Particular cases

Value of prime power $q$ Value of prime $p$ Exponent on $p$ giving $q$ Group order of the group = $q^3 - q$ subgroup structure page
2 2 1 symmetric group:S3 6 subgroup structure of symmetric group:S3
3 3 1 special linear group:SL(2,3) 24 subgroup structure of special linear group:SL(2,3)
4 2 2 alternating group:A5 60 subgroup structure of alternating group:A5
5 5 1 special linear group:SL(2,5) 120 subgroup structure of special linear group:SL(2,5)
7 7 1 special linear group:SL(2,7) 336 subgroup structure of special linear group:SL(2,7)
8 2 3 special linear group:SL(2,8) 504 subgroup structure of special linear group:SL(2,8)
9 3 2 special linear group:SL(2,9) 720 subgroup structure of special linear group:SL(2,9)

## Key statistics

Value of prime power $q$ Value of prime $p$ Exponent on $p$ giving $q$ Group order of the group = $q^3 - q$ Number of subgroups Number of conjugacy classes of subgroups Number of automorphism classes of subgroups Number of normal subgroups (= 3 for $q = 2$ or $q$ odd, $q \ge 5$, = 2 for $q$ a proper power of 2, = 4 for $q = 3$) Number of characteristic subgroups (equals number of normal subgroups in all cases)
2 2 1 symmetric group:S3 6 6 4 4 3 3
3 3 1 special linear group:SL(2,3) 24 15 7  ? 4 4
4 2 2 alternating group:A5 60 59 9 9 2 2
5 5 1 special linear group:SL(2,5) 120 76 12  ? 3 3
7 7 1 special linear group:SL(2,7) 336 224 19  ? 3 3
8 2 3 projective special linear group:PSL(2,8) 504 386 12  ? 2 2
9 3 2 special linear group:SL(2,9) 720 588 27  ? 3 3
11 11 1 special linear group:SL(2,11) 1320 766 21  ? 3 3
13 13 1 special linear group:SL(2,13) 2184 1140 21  ? 3 3
16 2 4 projective special linear group:PSL(2,16) 4080 3455 21  ? 2 2
17 17 1 special linear group:SL(2,17) 4896 2711 26  ? 3 3

## Sylow subgroups

We consider the group $SL(2,q)$ 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 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 Borel subgroup of degree two: $\{ \begin{pmatrix} a & b \\ 0 & a^{-1} \\\end{pmatrix} \mid a \in \mathbb{F}_q^\ast, b \in \mathbb{F}_q \}$
isomorphism class of $p$-Sylow normalizer It is the external semidirect product of $\mathbb{F}_q$ by the multiplicative group of $\mathbb{F}_q^\ast$ where the latter acts on the former via the multiplication action of the square of the acting element.
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:Z6 and for $q = 5$, it is dicyclic group:Dic20.
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 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 dicyclic 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 dicyclic group $2(q + 1)$ $q(q - 1)/2$
$\ell = 2$ and $q \equiv 1 \pmod 8$ dicyclic group (more specifically, a generalized quaternion group) dicyclic group (more specifically, a generalized quaternion 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 $q \equiv 5 \pmod 8$ quaternion group (special case of dicyclic group) special linear group:SL(2,3) 24 $(q^3 - q)/24$
$\ell = 2$ and $q \equiv 7 \pmod 8$ dicyclic group (more specifically, a generalized quaternion group) dicyclic group (more specifically, a generalized quaternion 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 $q \equiv 3 \pmod 8$ quaternion group (special case of dicyclic group) special linear group:SL(2,3) 24 $(q^3 - q)/24$

### Sylow subgroups for odd primes dividing $q - 1$

Suppose $\ell$ is an odd prime dividing $q - 1$. Note that $\ell \ne p$ and $\ell$ does not divide $q + 1$. Suppose $\ell^t$ is the largest power of $\ell$ dividing $q - 1$.

Item Value
order of $\ell$-Sylow subgroup $\ell^t$
index of $\ell$-Sylow subgroup $(q^3 - q)/\ell^t$
explicit description of one of the $\ell$-Sylow subgroups Since multiplicative group of a finite field is cyclic, $\mathbb{F}_q^\ast$ is cyclic of order $q - 1$. Let $H$ be its unique subgroup of order $\ell^t$. Then, the $\ell$-Sylow subgroup is $\{ \begin{pmatrix} a & 0 \\ 0 & a^{-1} \\\end{pmatrix} \mid a \in H \}$
isomorphism class of $\ell$-Sylow subgroup cyclic group of order $\ell^t$
explicit description of $\ell$-Sylow normalizer $\{ \begin{pmatrix} a & 0 \\ 0 & a^{-1} \\\end{pmatrix} \mid a \in \mathbb{F}_q^\ast \} \cup \{ \begin{pmatrix} 0 & a \\ -a^{-1} & 0 \\\end{pmatrix} \mid a \in \mathbb{F}_q^\ast \}$
isomorphism class of $\ell$-Sylow normalizer Case $p = 2$: dihedral group of order $2(q + 1)$
Case $p \ne 2$: dicyclic group of order $2(q + 1)$
order of $\ell$-Sylow normalizer $2(q - 1)$
$\ell$-Sylow number (i.e., number of $\ell$-Sylow subgroups) = index of $\ell$-Sylow normalizer $q(q + 1)/2$ (congruent to 1 mod $\ell$, as expected from the congruence condition on Sylow numbers)

### Sylow subgroups for odd primes dividing $q + 1$

Suppose $\ell$ is an odd prime dividing $q + 1$. Note that $\ell \ne p$ and $\ell$ does not divide $q - 1$. Suppose $\ell^t$ is the largest power of $\ell$ dividing $q + 1$.

Item Value
order of $\ell$-Sylow subgroup $\ell^t$
index of $\ell$-Sylow subgroup $(q^3 - q)/\ell^t$
explicit description of one of the $\ell$-Sylow subgroups Since multiplicative group of a finite field is cyclic, $\mathbb{F}_{q^2}^\ast$ is cyclic of order $q^2 - 1$. Further, via the action on a two-dimensional vector space over $\mathbb{F}_q$, we can embed $\mathbb{F}_{q^2}^\ast$ inside $GL(2,q)$. The image of the $\ell$-Sylow subgroup of $\mathbb{F}_{q^2}^\ast$ in $GL(2,q)$ actually lands inside $SL(2,q)$, and this image is a $\ell$-Sylow subgroup of $SL(2,q)$
isomorphism class of $\ell$-Sylow subgroup cyclic group of order $\ell^t$
explicit description of $\ell$-Sylow normalizer PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
isomorphism class of $\ell$-Sylow normalizer Case $p = 2$: dihedral group of order $2(q + 1)$
Case $p \ne 2$: dicyclic group of order $2(q + 1)$
order of $\ell$-Sylow normalizer $2(q + 1)$
$\ell$-Sylow number (i.e., number of $\ell$-Sylow subgroups) = index of $\ell$-Sylow normalizer $q(q - 1)/2$ (congruent to 1 mod $\ell$, as expected from the congruence condition on Sylow numbers)

### Sylow subgroups for the prime two where the field size is 1 mod 8

In this case, $(q + 1)/2$ is odd whereas $(q - 1)/2$ is even. In fact, $(q -1)/4$ is also even. Let $t$ be such that $2^t$ is the largest power of 2 dividing $q - 1$. Note that $t \ge 3$.

Item Value
order of 2-Sylow subgroup $2^{t+1}$
index of 2-Sylow subgroup $(q^3 - q)/2^{t+1}$
explicit description of one of the 2-Sylow subgroups Since multiplicative group of a finite field is cyclic, $\mathbb{F}_q^\ast$ is cyclic of order $q - 1$. Let $H$ be its unique subgroup of order $2^t$. Then, the 2-Sylow subgroup is $\{ \begin{pmatrix} a & 0 \\ 0 & a^{-1} \\\end{pmatrix} \mid a \in H \} \cup \{ \begin{pmatrix} 0 & a \\ -a^{-1} & 0 \\\end{pmatrix} \mid a \in H \}$
isomorphism class of 2-Sylow subgroup dicyclic group of order $2^{t + 1}$
explicit description of 2-Sylow normalizer Same as 2-Sylow subgroup
isomorphism class of 2-Sylow normalizer dicyclic group of order $2^{t + 1}$
order of 2-Sylow normalizer $2^{t + 1}$
2-Sylow number (i.e., number of 2-Sylow subgroups) = index of 2-Sylow normalizer $(q^3 - q)/2^{t + 1}$

### Sylow subgroups for the prime two where the field size is 5 mod 8

In this case, $(q + 1)/2$ is odd whereas $(q - 1)/2$ is even. However, $(q - 1)/4$ is odd. Then, $2^1$ is the largest power of 2 dividing $q + 1$ and $2^2$ is the largest power of 2 dividing $q - 1$.

Item Value
order of 2-Sylow subgroup 8
index of 2-Sylow subgroup $(q^3 - q)/8$
explicit description of one of the 2-Sylow subgroups Since multiplicative group of a finite field is cyclic, $\mathbb{F}_q^\ast$ is cyclic of order $q - 1$. Let $H$ be its unique subgroup of order 4. Then, the 2-Sylow subgroup is $\{ \begin{pmatrix} a & 0 \\ 0 & a^{-1} \\\end{pmatrix} \mid a \in H \} \cup \{ \begin{pmatrix} 0 & a \\ -a^{-1} & 0 \\\end{pmatrix} \mid a \in H \}$
isomorphism class of 2-Sylow subgroup quaternion group
explicit description of 2-Sylow normalizer PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
isomorphism class of 2-Sylow normalizer special linear group:SL(2,3)
order of 2-Sylow normalizer 24
2-Sylow number (i.e., number of 2-Sylow subgroups) = index of 2-Sylow normalizer $(q^3 - q)/24$

### Sylow subgroups for the prime two where the field size is 7 mod 8

In this case, $(q - 1)/2$ is odd whereas $(q + 1)/2$ is even. Let $t$ be such that $2^t$ is the largest power of 2 dividing $q + 1$. Note that $t \ge 3$.

Item Value
order of 2-Sylow subgroup $2^{t+1}$
index of 2-Sylow subgroup $(q^3 - q)/2^{t+1}$
explicit description of one of the 2-Sylow subgroups PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
isomorphism class of 2-Sylow subgroup dicyclic group of order $2^{t + 1}$
explicit description of 2-Sylow normalizer Same as 2-Sylow subgroup
isomorphism class of 2-Sylow normalizer dicyclic group of order $2^{t + 1}$
order of 2-Sylow normalizer $2^{t + 1}$
2-Sylow number (i.e., number of 2-Sylow subgroups) = index of 2-Sylow normalizer $(q^3 - q)/2^{t + 1}$

### Sylow subgroups for the prime two where the field size is 3 mod 8

In this case, $(q - 1)/2$ is odd whereas $(q + 1)/2$ is even, but $(q + 1)/4$ is odd. Then, the largest power of 2 dividing $q - 1$ is 2 and the largest power of 2 dividing $q + 1$ is 4.

Item Value
order of 2-Sylow subgroup 8
index of 2-Sylow subgroup $(q^3 - q)/8$
explicit description of one of the 2-Sylow subgroups PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
isomorphism class of 2-Sylow subgroup quaternion group (dicyclic group of order 8)
explicit description of 2-Sylow normalizer PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
isomorphism class of 2-Sylow normalizer special linear group:SL(2,3)
order of 2-Sylow normalizer 24
2-Sylow number (i.e., number of 2-Sylow subgroups) = index of 2-Sylow normalizer $(q^3 - q)/24$