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 | Value of prime | Exponent on giving | Group | order of the group = | 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 | Value of prime | Exponent on giving | Group | order of the group = | Number of subgroups | Number of conjugacy classes of subgroups | Number of automorphism classes of subgroups | Number of normal subgroups (= 3 for or odd, , = 2 for a proper power of 2, = 4 for ) | 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 over the field of elements. is a prime power of the form where is a prime number and is a positive integer. is hence also the characteristic of . We call the characteristic prime.
Sylow subgroups for the characteristic prime
Item | Value |
---|---|
order of -Sylow subgroup | |
index of -Sylow subgroup | |
explicit description of one of the -Sylow subgroups | unitriangular matrix group of degree two: |
isomorphism class of -Sylow subgroup | additive group of , which is an elementary abelian group of order , i.e., a direct product of copies of the cyclic group of order |
explicit description of -Sylow normalizer | Borel subgroup of degree two: |
isomorphism class of -Sylow normalizer | It is the external semidirect product of by the multiplicative group of where the latter acts on the former via the multiplication action of the square of the acting element. For (so ), it is isomorphic to the general affine group of degree one . For , it is cyclic group:Z6 and for , it is dicyclic group:Dic20. |
order of -Sylow normalizer | |
-Sylow number (i.e., number of -Sylow subgroups) = index of -Sylow normalizer | (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 , the -Sylow subgroup is nontrivial iff . If , then it does not divide , so we get that which means that either or . Further, if , exactly one of these cases can occur. For , we make cases based on the residue of mod 8. The summary of cases is below and more details are in later sections.
Case on and | Isomorphism type of -Sylow subgroup | Isomorphism type of -Sylow normalizer | Order of -Sylow normalizer | -Sylow number = index of -Sylow normalizer |
---|---|---|---|---|
is an odd prime dividing , | cyclic group | dihedral group | ||
is an odd prime dividing , | cyclic group | dicyclic group | ||
is an odd prime dividing , | cyclic group | dihedral group | ||
is an odd prime dividing , | cyclic group | dicyclic group | ||
and | 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 | largest odd number dividing the order |
and | quaternion group (special case of dicyclic group) | special linear group:SL(2,3) | 24 | |
and | 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 | largest odd number dividing the order |
and | quaternion group (special case of dicyclic group) | special linear group:SL(2,3) | 24 |
Sylow subgroups for odd primes dividing
Suppose is an odd prime dividing . Note that and does not divide . Suppose is the largest power of dividing .
Item | Value |
---|---|
order of -Sylow subgroup | |
index of -Sylow subgroup | |
explicit description of one of the -Sylow subgroups | Since multiplicative group of a finite field is cyclic, is cyclic of order . Let be its unique subgroup of order . Then, the -Sylow subgroup is |
isomorphism class of -Sylow subgroup | cyclic group of order |
explicit description of -Sylow normalizer | |
isomorphism class of -Sylow normalizer | Case : dihedral group of order Case : dicyclic group of order |
order of -Sylow normalizer | |
-Sylow number (i.e., number of -Sylow subgroups) = index of -Sylow normalizer | (congruent to 1 mod , as expected from the congruence condition on Sylow numbers) |
Sylow subgroups for odd primes dividing
Suppose is an odd prime dividing . Note that and does not divide . Suppose is the largest power of dividing .
Item | Value |
---|---|
order of -Sylow subgroup | |
index of -Sylow subgroup | |
explicit description of one of the -Sylow subgroups | Since multiplicative group of a finite field is cyclic, is cyclic of order . Further, via the action on a two-dimensional vector space over , we can embed inside . The image of the -Sylow subgroup of in actually lands inside , and this image is a -Sylow subgroup of |
isomorphism class of -Sylow subgroup | cyclic group of order |
explicit description of -Sylow normalizer | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
isomorphism class of -Sylow normalizer | Case : dihedral group of order Case : dicyclic group of order |
order of -Sylow normalizer | |
-Sylow number (i.e., number of -Sylow subgroups) = index of -Sylow normalizer | (congruent to 1 mod , 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, is odd whereas is even. In fact, is also even. Let be such that is the largest power of 2 dividing . Note that .
Item | Value |
---|---|
order of 2-Sylow subgroup | |
index of 2-Sylow subgroup | |
explicit description of one of the 2-Sylow subgroups | Since multiplicative group of a finite field is cyclic, is cyclic of order . Let be its unique subgroup of order . Then, the 2-Sylow subgroup is |
isomorphism class of 2-Sylow subgroup | dicyclic group of order |
explicit description of 2-Sylow normalizer | Same as 2-Sylow subgroup |
isomorphism class of 2-Sylow normalizer | dicyclic group of order |
order of 2-Sylow normalizer | |
2-Sylow number (i.e., number of 2-Sylow subgroups) = index of 2-Sylow normalizer |
Sylow subgroups for the prime two where the field size is 5 mod 8
In this case, is odd whereas is even. However, is odd. Then, is the largest power of 2 dividing and is the largest power of 2 dividing .
Item | Value |
---|---|
order of 2-Sylow subgroup | 8 |
index of 2-Sylow subgroup | |
explicit description of one of the 2-Sylow subgroups | Since multiplicative group of a finite field is cyclic, is cyclic of order . Let be its unique subgroup of order 4. Then, the 2-Sylow subgroup is |
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 |
Sylow subgroups for the prime two where the field size is 7 mod 8
In this case, is odd whereas is even. Let be such that is the largest power of 2 dividing . Note that .
Item | Value |
---|---|
order of 2-Sylow subgroup | |
index of 2-Sylow subgroup | |
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 |
explicit description of 2-Sylow normalizer | Same as 2-Sylow subgroup |
isomorphism class of 2-Sylow normalizer | dicyclic group of order |
order of 2-Sylow normalizer | |
2-Sylow number (i.e., number of 2-Sylow subgroups) = index of 2-Sylow normalizer |
Sylow subgroups for the prime two where the field size is 3 mod 8
In this case, is odd whereas is even, but is odd. Then, the largest power of 2 dividing is 2 and the largest power of 2 dividing is 4.
Item | Value |
---|---|
order of 2-Sylow subgroup | 8 |
index of 2-Sylow subgroup | |
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 |