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, $q\geq 5$, = 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 $\ell \neq p$, 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 $\ell \neq 2$, 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 $q-1$, $p\neq 2$	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 $q+1$, $p\neq 2$	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 $\ell \neq p$ 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 $p\neq 2$: 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 $\ell \neq p$ 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] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
isomorphism class of $\ell$-Sylow normalizer	Case $p=2$: dihedral group of order $2(q+1)$ Case $p\neq 2$: 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 $t\geq 3$.

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] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
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 $t\geq 3$.

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] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
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] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
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] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
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	$(q^{3}-q)/24$