2-Sylow subgroup 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) quaternion group 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 cyclic group:Z3.
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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 $2\times 2$ matrices of determinant 1 over the field with three elements. The field has elements 0,1,2, with $2=-1$.

$H$ is the subgroup:

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

$H$ is a normal subgroup of $G$ and is isomorphic to the quaternion group of order 8.

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	8
index of the subgroup	3
size of conjugacy class = 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	$\{{\begin{pmatrix}1&0\\0&1\\\end{pmatrix}},{\begin{pmatrix}2&0\\0&2\\\end{pmatrix}}\}$	center of special linear group:SL(2,3)	cyclic group:Z2
normal core	the subgroup itself	current page	quaternion group
normal closure	the subgroup itself	current page	quaternion group
characteristic core	the subgroup itself	current page	quaternion group
characteristic closure	the subgroup itself	current page	quaternion group
commutator with whole group	the subgroup itself	current page	quaternion group

Subgroup-defining functions

The subgroup is a characteristic subgroup of the whole group and arises as the result of many subgroup-defining functions. Some of these are given below.

Subgroup-defining function	Meaning in general	Why it takes this value
derived subgroup	subgroup generated by commutators of all pairs of group elements, smallest subgroup with abelian quotient	The quotient is cyclic group:Z3, which is abelian; no other subgroup has abelian quotient. We can also explicitly compute all commutators and obtain all the elements of the subgroup.
Fitting subgroup	join of all nilpotent normal subgroups	The only nilpotent normal subgroups of the whole group are this subgroup, the trivial subgroup, and the center of the whole group (which is also the center of the subgroup).
2-Sylow core	largest normal subgroup whose order is a power of 2; normal core of any 2-Sylow subgroup	The subgroup is the unique normal 2-Sylow subgroup
2-Sylow closure	normal closure of any 2-Sylow subgroup	The subgroup is the unique normal 2-Sylow subgroup
Jacobson radical	intersection of all maximal normal subgroups	The subgroup is the unique maximal normal subgroup -- the group is a one-headed group