Center of special linear group:SL(2,3)
From Groupprops
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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Contents
Definition
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 .
is the subgroup:
is isomorphic to cyclic group:Z2. It is the center of . The quotient group is the projective special linear group of degree two over field:F3, which is isomorphic to alternating group:A4.
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of the whole group | 24 | order of is . Here . |
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. |