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

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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