# Center of special linear group:SL(2,5)

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(2,5) (see subgroup structure of special linear group:SL(2,5)).
The subgroup is a normal subgroup and the quotient group is isomorphic to alternating group:A5.
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## Definition $G$ is the special linear group:SL(2,5), i.e., the special linear group of degree two over field:F5. In other words, it is the group of invertible matrices of determinant 1 over the field with three elements. The field has elements 0,1,2,3,4 with $4 = -1$. $H$ is the subgroup: $\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}, \begin{pmatrix} 4 & 0 \\ 0 & 4 \\\end{pmatrix} \}$ $H$ 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 $PSL(2,5)$ over field:F5, which is isomorphic to alternating group:A5.

## Arithmetic functions

Function Value Explanation
order of the whole group 120 order of $SL(2,q)$ is $q^3 - q$. Here $q = 5$.
order of the subgroup 2
index of the subgroup 60
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(2,5)
centralizer the whole group -- special linear group:SL(2,5)
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
center commutes with all group elements
socle join of all minimal normal subgroups It is the unique minimal normal subgroup, i.e., the whole group is a monolithic group.
Jacobson radical intersection of all maximal normal subgroups It is the unique maximal normal subgroup, i.e., the whole group is a one-headed group.

## 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 verbal and from being homomorph-containing, see below.
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 $30^{th}$ 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.