# SL(2,3) in GL(2,3)

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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## Definition

The group $G$ is general linear group:GL(2,3): the general linear group of degree two over field:F3, i.e., the group of all invertible $2 \times 2$ matrices over the field, with matrix multiplication:

$\! G := \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in \mathbb{F}_3, ad \ne bc \}$

The subgroup $H$ 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 $1$.

$\! H := \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in \mathbb{F}_3, ad - bc = 1 \}$

Equivalently, $H$ is the kernel of the determinant homomorphism from $G$ to the multiplicative group of $\mathbb{F}_3$, i.e., the map:

$\begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mapsto ad - bc$

## Arithmetic functions

Function Value Explanation
order of whole group 48 As $GL(2,q)$, $q = 3$: $q(q+1)(q-1)^2 = 3(4)(2)^2 = 48$
order of subgroup 24 As $SL(2,q)$, $q = 3$: $q(q+1)(q-1) = q^3 - q = 3(4)(2) = 3^3 - 3 = 24$
index of subgroup 2 Using $q = 3$: order of multiplicative group $\mathbb{F}_q^\ast$, which is $q - 1 = 3 - 1 = 2$

## 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)
$\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$, $\begin{pmatrix} 2 & 0 \\ 0 & 2 \\\end{pmatrix}$ cyclic group:Z2 center of special linear group:SL(2,3) center of general linear group:GL(2,3) alternating group:A4 (considered as $PSL(2,3)$) symmetric group:S4 (considered as $PGL(2,3)$) 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 $GL(2,2)$) 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

### Advanced properties related to resemblance and corollaries for invariance

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 $H$ are (up to isomorphism), SL(2,3), cyclic group:Z3, and alternating group:A4 -- none of these are isomorphic to subgroups of $G$.
subhomomorph-containing subgroup contains every homomorphic image of every subgroup No The subgroup quaternion group has a homomorphic image of order two generated by $\begin{pmatrix} 1 & 0 \\ 0 & -1 \\\end{pmatrix}$.

### 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 $G$ is given as a black-box group (i.e., not explicitly defined as $GL(2,3)$), $H$ can be constructed as its derived subgroup:

H = DerivedSubgroup(G);