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.