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

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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]

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]
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 $q(q - 1) = p^{2r} - p^r$ 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]
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]
isomorphism class of $\ell$-Sylow normalizer Case $p = 2$: dihedral group of order $2(q + 1)$
Case $p \ne 2$: dicyclic group of order $2(q + 1)$
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