View source for 2-core of general linear group:GL(2,3)

{{particular subgroup|
subgroup = quaternion group|
group = general linear group:GL(2,3)|
quotient group = symmetric group:S3}}

==Definition==

<math>G</math> is the [[general linear group:GL(2,3)]], i.e., the [[general linear group]] of [[general linear group of degree two|degree two]] over [[field:F3]]. In other words, it is the group of invertible <math>2 \times 2</math> matrices over the field with three elements. The field has elements 0,1,2, with <math>2 = -1</math>.

<math>H</math> is the subgroup:

<math>\{ \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} \}</math>

<math>H</math> is a [[normal subgroup]] of <math>G</math> and is isomorphic to the [[quaternion group]] of order 8. The [[quotient group]] is isomorphic to [[symmetric group:S3]].

==Arithmetic functions==

{| class="sortable" border="1"
! Function !! Value !! Explanation
|-
| [[order of a group|order]] of the whole group || 48 || Order of <math>GL(2,q)</math> is <math>(q^2 - q)(q^2 - 1)</math>. Here <math>q = 3</math>.
|-
| [[order of a group|order]] of the subgroup || 8 ||
|-
| [[index of a subgroup|index]] of the subgroup || [[arithmetic function value::index of a subgroup;6|6]] ||
|-
| size of conjugacy class = index of normalizer || 1 ||
|-
| number of conjugacy classes in automorphism class || 1 ||
|}

==Effect of subgroup operators==

{| class="sortable" border="1"
! Function !! Value as subgroup (descriptive) !! Value as subgroup (link) !! Value as group
|-
| [[normalizer]] || the whole group || -- || [[general linear group:GL(2,3)]]
|-
| [[centralizer]] || <math>\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}, \begin{pmatrix} 2 & 0 \\ 0 & 2 \\\end{pmatrix}\}</math> || [[center of general 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 function]]s. Some of these are given below.

{| class="sortable" border="1"
! Subgroup-defining function !! Meaning in general !! Why it takes this value
|-
| [[arises as subgroup-defining function::second derived subgroup]] || [[derived subgroup]] of [[derived subgroup]] || The derived subgroup is [[special linear group:SL(2,3)]] (see [[SL(2,3) in GL(2,3)]]) and the derived subgroup of that is this subgroup (see [[Q8 in SL(2,3)]]).
|-
| [[arises as subgroup-defining function::Fitting subgroup]] || join of all [[nilpotent normal subgroup]]s || 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-[[arises as subgroup-defining function::Sylow core]] or 2-core || largest normal subgroup whose order is a power of 2; [[normal core]] of any 2-Sylow subgroup || There are three 2-Sylow subgroups, each of order 16 and isomorphic to [[semidihedral group:SD16]] (See [[SD16 in GL(2,3)]]). Their intersection is precisely this subgroup.
|}