SL(2,3) in GL(2,3)
From Groupprops
This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) special linear group:SL(2,3) and the group is (up to isomorphism) general linear group:GL(2,3) (see subgroup structure of general linear group:GL(2,3)).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2.
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Contents
Definition
The group is general linear group:GL(2,3): the general linear group of degree two over field:F3, i.e., the group of all invertible matrices over the field, with matrix multiplication:
The subgroup is special linear group:SL(2,3): the special linear group of degree two over field:F3, i.e., the subgroup comprising matrices of determinant .
Equivalently, is the kernel of the determinant homomorphism from to the multiplicative group of , i.e., the map:
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of whole group | 48 | As , : |
order of subgroup | 24 | As , : |
index of subgroup | 2 | Using : order of multiplicative group , which is |
Related subgroups
Smaller normal subgroups
Description of smaller subgroup | Isomorphism class of smaller subgroup | Smaller subgroup in subgroup | Smaller subgroup in group | Quotient of subgroup by smaller subgroup | Quotient of group by smaller subgroup | (Quotient of subgroup by smaller subgroup) as subgroup of (quotient of group by smaller subgroup) |
---|---|---|---|---|---|---|
, | cyclic group:Z2 | center of special linear group:SL(2,3) | center of general linear group:GL(2,3) | alternating group:A4 (considered as ) | symmetric group:S4 (considered as ) | A4 in S4 |
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] | quaternion group | Q8 in SL(2,3) | Q8 in GL(2,3) | cyclic group:Z3 | symmetric group:S3 | A3 in S3 |
Subgroup-defining functions
Subgroup-defining function | Meaning in general | Why it takes this value | GAP verification (set G := GL(2,3); H := SL(2,3);) |
---|---|---|---|
derived subgroup (also called commutator subgroup) | subgroup generated by all commutators | derived subgroup of general linear group is special linear group (there is an exception for ) | H = DerivedSubgroup(G); using DerivedSubgroup |
Jacobson radical | unique maximal normal subgroup |
Subgroup properties
Invariance under automorphisms and endomorphisms: basic properties
Property | Meaning | Satisfied? | Explanation | GAP verification (set G := GL(2,3); H := SL(2,3);) |
---|---|---|---|---|
normal subgroup | invariant under all inner automorphisms | Yes | Follows from index two implies normal, among other reasons | IsNormal(G,H); using IsNormal |
characteristic subgroup | invariant under all automorphisms | Yes | derived subgroup is characteristic | IsCharacteristicSubgroup(G,H); using IsCharacteristicSubgroup |
fully invariant subgroup | invariant under all endomorphisms | Yes | derived subgroup is fully invariant | IsFullinvariant(G,H); using IsFullinvariant |
Property | Meaning | Satisfied? | Explanation | GAP verification (set G := GL(2,3); H := SL(2,3);) |
---|---|---|---|---|
order-unique subgroup, index-unique subgroup | unique subgroup of its order (respectively index). Note that order-unique and index-unique are equivalent for subgroups of finite groups | Yes | ||
isomorph-free subgroup | no other isomorphic subgroup | Yes | Follows from being order-unique | |
isomorph-containing subgroup | contains every isomorphic subgroup (this property is equivalent to being isomorph-free for finite subgroups) | Yes | Follows from being isomorph-free | |
quotient-isomorph-free subgroup | normal, and no other normal subgroup with an isomorphic quotient group | Yes | Follows from being index-unique | |
homomorph-containing subgroup | contains every homomorphic image in whole group | Yes | The only nontrivial quotient groups of are (up to isomorphism), SL(2,3), cyclic group:Z3, and alternating group:A4 -- none of these are isomorphic to subgroups of . | |
subhomomorph-containing subgroup | contains every homomorphic image of every subgroup | No | The subgroup quaternion group has a homomorphic image of order two generated by . |
Transitivity of normality
Property | Meaning | Satisfied? | Explanation | GAP verification (set G := GL(2,3); H := SL(2,3);) |
---|---|---|---|---|
transitively normal subgroup | every normal subgroup of the subgroup is normal in the whole group. | Yes | ? | |
central factor | product with centralizer is whole group | No | centralizer of subgroup is same as center of whole group and of subgroup (see center of SL(2,3), center of GL(2,3)) | |
self-centralizing subgroup | contains its centralizer in whole group | Yes | centralizer of subgroup is center of SL(2,3) |
GAP implementation
Direct construction
The group-subgroup pair can be constructed using GL and SL:
G := GL(2,3); H := SL(2,3);
Using a black-box group
If is given as a black-box group (i.e., not explicitly defined as ), can be constructed as its derived subgroup:
H = DerivedSubgroup(G);