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

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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]


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.
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
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