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] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	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);