# Center of special linear group:SL(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) cyclic group:Z2 and the group is (up to isomorphism) special linear group:SL(2,3) (see subgroup structure of special linear group:SL(2,3)).
The subgroup is a normal subgroup and the quotient group is isomorphic to alternating group:A4.
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## Definition

$G$ is the special linear group:SL(2,3), i.e., the special linear group of degree two over field:F3. In other words, it is the group of invertible matrices of determinant 1 over the field with three elements. The field has elements 0,1,2, with $2 = -1$.

$H$ is the subgroup:

$\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}, \begin{pmatrix} 2 & 0 \\ 0 & 2 \\\end{pmatrix} \}$

$H$ is isomorphic to cyclic group:Z2. It is the center of $G$. The quotient group $G/H$ is the projective special linear group of degree two $PSL(2,3)$ over field:F3, which is isomorphic to alternating group:A4.

## Arithmetic functions

Function Value Explanation
order of the whole group 24 order of $SL(2,q)$ is $q^3 - q$. Here $q = 3$.
order of the subgroup 2
index of the subgroup 12
size of conjugacy class of subgroup = index of normalizer 1
number of conjugacy classes in automorphism class 1

## Effect of subgroup operators

Function Value as subgroup (descriptive) Value as subgroup (link) Value as group
normalizer the whole group -- special linear group:SL(2,3)
centralizer the whole group -- special linear group:SL(2,3)
normal core the subgroup itself current page cyclic group:Z2
normal closure the subgroup itself current page cyclic group:Z2
characteristic core the subgroup itself current page cyclic group:Z2
characteristic closure the subgroup itself current page cyclic group:Z2
commutator with whole group trivial subgroup current page trivial group

## Subgroup-defining functions

The subgroup is a characteristic subgroup and arises as a result of many subgroup-defining functions, some of which are detailed below:

Subgroup-defining function Meaning in general Why it takes this value
second derived subgroup derived subgroup of derived subgroup The derived subgroup is 2-Sylow subgroup of special linear group:SL(2,3), which is isomorphic to quaternion group, and this is its center.
socle join of all minimal normal subgroups It is the unique minimal normal subgroup, i.e., the whole group is a monolithic group.
center commutes with all group elements

## Subgroup properties

### Invariance under automorphisms and endomorphisms

Property Meaning Satisfied? Explanation Comment
normal subgroup invariant under inner automorphisms Yes center is normal
characteristic subgroup invariant under all automorphisms Yes center is characteristic
fully invariant subgroup invariant under all endomorphisms Yes Follows from being a member of the derived series (here, the second derived subgroup). Also from its being the set of elements of order dividing 2.
homomorph-containing subgroup contains image under any homomorphism to the group Yes Follows from being precisely the set of elements of order dividing 2.
verbal subgroup generated by set of words Yes Precisely the set of sixth powers in the group
1-endomorphism-invariant subgroup invariant under all 1-endomorphisms Yes Follows from being precisely the set of elements of order dividing 2.
1-automorphism-invariant subgroup invariant under all 1-automorphisms Yes Follows from being 1-endomorphism-invariant.
quasiautomorphism-invariant subgroup invariant under all quasiautomorphisms Yes Follows from being 1-automorphism-invariant.