Subgroup structure of special linear group:SL(2,5)
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This article gives specific information, namely, subgroup structure, about a particular group, namely: special linear group:SL(2,5).
View subgroup structure of particular groups | View other specific information about special linear group:SL(2,5)
This article describes the subgroup structure of special linear group:SL(2,5), which is the special linear group of degree two over field:F5. The group has order 120.
Family contexts
Family name | Parameter values | General discussion of subgroup structure of family |
---|---|---|
special linear group of degree two | field:F5, i.e., the group | subgroup structure of special linear group of degree two over a finite field |
double cover of alternating group | degree , i.e., the group | 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 | 76 |
number of conjugacy classes of subgroups | 12 |
number of automorphism classes of subgroups | 12 |
isomorphism classes of Sylow subgroups, corresponding fusion systems, and Sylow numbers | 2-Sylow: quaternion group (order 8) with its non-inner fusion system (see non-inner fusion system for quaternion group), Sylow number 5 3-Sylow: cyclic group:Z3 with its non-inner fusion system, Sylow number 10 5-Sylow: cyclic group:Z5, Sylow number 6 |
Hall subgroups | Other than the whole group, trivial subgroup, and Sylow subgroups, there is a -Hall subgroup of order 24 (SL(2,3) in SL(2,5)). There is no -Hall subgroup or -Hall subgroup. |
maximal subgroups | There are maximal subgroups of order 12 (index 10), order 20 (index 6) and order 24 (index 5). |
normal subgroups | The only proper nontrivial normal subgroup is center of special linear group:SL(2,5), which is isomorphic to cyclic group:Z2 and the quotient group is isomorphic to alternating group:A5. |
Defining functions
Subgroup-defining functions and associated quotient-defining functions
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 with a prime power, . The prime is the characteristic prime.
Sylow subgroups for the prime 5
The prime 5 is the characteristic prime , so we compare with the general information on -Sylow subgroups of .
Item | Value for , generic | Value for (so ) |
---|---|---|
order of -Sylow subgroup | 5 | |
index of -Sylow subgroup | 24 | |
explicit description of one of the -Sylow subgroups | unitriangular matrix group of degree two: | See 5-Sylow subgroup of special linear group:SL(2,5) |
isomorphism class of -Sylow subgroup | additive group of the field , which is an elementary abelian group of order | cyclic group:Z5 |
explicit description of -Sylow normalizer | Borel subgroup of degree two: | See 5-Sylow normalizer of special linear group:SL(2,5) |
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. |
dicyclic group:Dic20 |
order of -Sylow normalizer | 20 | |
-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) | 6 |
Sylow subgroups for the prime 2
We are in the subcase where ( being the prime for which we are taking Sylow subgroups) and .
Item | Value for | Value for (our case) |
---|---|---|
order of 2-Sylow subgroup | 8 | 8 |
index of 2-Sylow subgroup | 15 | |
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 | 2-Sylow subgroup of special linear group:SL(2,5) |
isomorphism class of 2-Sylow subgroup | quaternion group | quaternion group |
explicit description of 2-Sylow normalizer | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] | SL(2,3) in SL(2,5) |
isomorphism class of 2-Sylow normalizer | special linear group:SL(2,3) | special linear group:SL(2,3) |
order of 2-Sylow normalizer | 24 | 24 |
2-Sylow number (i.e., number of 2-Sylow subgroups) = index of 2-Sylow normalizer | 5 |
Sylow subgroups for the prime 3
Here, and we are interested in the -Sylow subgroups.
We are in the subcase is an odd prime dividing . Suppose is the largest power of dividing . In our case, .
Item | Value for generic | Value for |
---|---|---|
order of -Sylow subgroup | 3 | |
index of -Sylow subgroup | 40 | |
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 | 3-Sylow subgroup of special linear group:SL(2,5) |
isomorphism class of -Sylow subgroup | cyclic group of order | cyclic group:Z3 |
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 |
dicyclic group:Dic12 |
order of -Sylow normalizer | 12 | |
-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) | 10 |