Subgroup structure of special linear group:SL(2,9)

This article gives specific information, namely, subgroup structure, about a particular group, namely: special linear group:SL(2,9).
View subgroup structure of particular groups | View other specific information about special linear group:SL(2,9)

This article describes the subgroup structure of special linear group:SL(2,9), which is the special linear group of degree two over field:F9. The group has order 720.

Family contexts

Family name	Parameter values	General discussion of subgroup structure of family
special linear group of degree two	field:F9, i.e., the group $SL(2,9)$	subgroup structure of special linear group of degree two over a finite field
double cover of alternating group $2\cdot A_{n}$	degree $n=6$, i.e., the group $2\cdot A_{6}$	subgroup structure of double cover of alternating group

Tables for quick information

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
normal Hall implies permutably complemented, Hall retract implies order-conjugate

Quick summary

Item	Value
number of subgroups	588
number of conjugacy classes of subgroups	27
number of automorphism classes of 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.

Sylow subgroups

Compare and contrast with subgroup structure of special linear group of degree two over a finite field#Sylow subgroups

We are considering the group $SL(2,q)$ with $q=p^{r}$ a prime power, $q=9,p=3,r=2$. The prime $p=3$ is the characteristic prime.

Sylow subgroups for the prime 3

The prime 3 is the characteristic prime $p$, so we compare with the general information on $p$-Sylow subgroups of $SL(2,q)$.

Item	Value for $SL(2,q)$, generic $q$	Value for $SL(2,9)$ (so $q=9,p=3,r=2$)
order of $p$-Sylow subgroup	$q$	9
index of $p$-Sylow subgroup	$q^{2}-1$	80
explicit description of one of the $p$-Sylow subgroups	unitriangular matrix group of degree two: $\{{\begin{pmatrix}1&b\\0&1\\\end{pmatrix}}\mid b\in \mathbb {F} _{q}\}$	See 3-Sylow subgroup of special linear group:SL(2,9)
isomorphism class of $p$-Sylow subgroup	additive group of the field $\mathbb {F} _{q}$, which is an elementary abelian group of order $p^{r}$	elementary abelian group:E9
explicit description of $p$-Sylow normalizer	Borel subgroup of degree two: $\{{\begin{pmatrix}a&b\\0&a^{-1}\\\end{pmatrix}}\mid a\in \mathbb {F} _{q}^{\ast },b\in \mathbb {F} _{q}\}$	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 $p$-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.	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.
order of $p$-Sylow normalizer	$q(q-1)=p^{2r}-p^{r}$	72
$p$-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)	10

Sylow subgroups for the prime 2

We are in the subcase where $\ell =2$ ($\ell$ being the prime for which we are taking Sylow subgroups) and $q\equiv 1{\pmod {8}}$. The value $t$ such that $2^{t}$ is the largest power of 2 dividing $q-1$ is $t=3$.

Item	Value for $\ell =2,q\equiv 1{\pmod {8}}$	Value for $\ell =2,t=3,q=9,p=3,r=2$ (our case)
order of 2-Sylow subgroup	$2^{t+1}$	16
index of 2-Sylow subgroup	$(q^{3}-q)/2^{t+1}$	45
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 $\{{\begin{pmatrix}a&0\\0&a^{-1}\\\end{pmatrix}}\mid a\in H\}\cup \{{\begin{pmatrix}0&a\\-a^{-1}&0\\\end{pmatrix}}\mid a\in H\}$	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 $2^{t+1}$	generalized quaternion group:Q16
explicit description of 2-Sylow normalizer	Same as 2-Sylow subgroup	Same as 2-Sylow subgroup
isomorphism class of 2-Sylow normalizer	dicyclic group of order $2^{t+1}$	generalized quaternion group:Q16
order of 2-Sylow normalizer	$2^{t+1}$	16
2-Sylow number (i.e., number of 2-Sylow subgroups) = index of 2-Sylow normalizer	$(q^{3}-q)/2^{t+1}$	45

Sylow subgroups for the prime 5

Here, $\ell =5$ and we are interested in the $\ell$-Sylow subgroups.

We are in the subcase $\ell$ is an odd prime dividing $q+1$. Suppose $\ell ^{t}$ is the largest power of $\ell$ dividing $q+1$. In our case, $t=1$.

Item	Value for generic $\ell ,t,p,q,r$	Value for $\ell =5,t=1,q=9,p=3,r=2$
order of $\ell$-Sylow subgroup	$\ell ^{t}$	5
index of $\ell$-Sylow subgroup	$(q^{3}-q)/\ell ^{t}$	144
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 $SL(2,q)$	5-Sylow subgroup of special linear group:SL(2,9)
isomorphism class of $\ell$-Sylow subgroup	cyclic group of order $\ell ^{t}$	cyclic group:Z5
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 $2(q+1)$	dicyclic group:Dic20
order of $\ell$-Sylow normalizer	$2(q+1)$	20
$\ell$-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)	36