# 2-Sylow subgroup of general linear group:GL(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) semidihedral group:SD16 and the group is (up to isomorphism) general linear group:GL(2,3) (see subgroup structure of general linear group:GL(2,3)).
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## Definition $G$ is the general linear group of degree two over field:F3. In other words, it is the group of invertible $2 \times 2$ matrices with entries over the field of three elements. The field has elements $0,1,2$ with $2 = -1$. $H$ is one of the 2-Sylow subgroups of $G$, i.e.: $H$ is isomorphic to semidihedral group:SD16.

## Arithmetic functions

Function Value Explanation
order of the whole group 48
order of the subgroup 16
index of the subgroup 3
size of conjugacy class of subgroups 3
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 subgroup itself current page semidihedral group:SD16
centralizer $\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}, \begin{pmatrix} 2 & 0 \\ 0 & 2 \\\end{pmatrix}$ center of general linear group:GL(2,3) cyclic group:Z2
normal core $\langle \begin{pmatrix} 0 & 1 \\ 2 & 0 \\\end{pmatrix}, \begin{pmatrix} 2 & 2 \\ 2 & 1 \\\end{pmatrix}$ 2-core of general linear group:GL(2,3) quaternion group
normal closure the whole group -- general linear group:GL(2,3)
characteristic core $\langle \begin{pmatrix} 0 & 1 \\ 2 & 0 \\\end{pmatrix}, \begin{pmatrix} 2 & 2 \\ 2 & 1 \\\end{pmatrix}$ 2-core of general linear group:GL(2,3) quaternion group
characteristic closure the whole group -- general linear group:GL(2,3)
commutator with whole group all matrices of determinant 1 SL(2,3) in GL(2,3) special linear group:SL(2,3)

## Conjugacy class-defining functions

Conjugacy class-defining function Meaning in general Why it takes this value
2-Sylow subgroups $p$-Sylow subgroups are subgroups whose order is a power of $p$ and index is relatively prime to $p$. Sylow subgroups exist and Sylow implies order-conjugate, i.e., any two $p$-Sylow subgroups are conjugate. Here $p = 2$. The subgroups have order $16 = 2^4$ and index 3.
2-Sylow normalizers Normalizer of a 2-Sylow subgroup The 2-Sylow subgroups are self-normalizing subgroups.

## Subgroup properties

### Sylow and corollaries

The subgroup is a 3-Sylow subgroup, so many properties follow as a corollary of that.

Property Meaning Why being Sylow implies the property
order-conjugate subgroup conjugate to any subgroup of the same order Sylow implies order-conjugate
isomorph-conjugate subgroup conjugate to any subgroup isomorphic to it (via order-conjugate)
automorph-conjugate subgroup conjugate to any subgroup automorphic to it (via isomorph-conjugate)
intermediately isomorph-conjugate subgroup conjugate to any subgroup isomorphic to it inside any intermediate subgroup Sylow implies intermediately isomorph-conjugate
intermediately automorph-conjugate subgroup automorph-conjugate in every intermediate subgroup (via intermediately isomorph-conjugate)
pronormal subgroup conjugate to any conjugate subgroup in their join Sylow implies pronormal
weakly pronormal subgroup (via pronormal)
paranormal subgroup (via pronormal)
polynormal subgroup (via pronormal)

### Normality-related properties

Property Meaning Satisfied? Explanation Comment
normal subgroup equals all its conjugate subgroups No (see above for other conjugate subgroups)
2-subnormal subgroup normal subgroup in its normal closure No
subnormal subgroup series from subgroup to whole group, each normal in next No
contranormal subgroup normal closure is whole group Yes
self-normalizing subgroup equals its normalizer in the whole group Yes
abnormal subgroup Yes Sylow normalizer implies abnormal
weakly abnormal subgroup Yes (via abnormal)
self-centralizing subgroup contains its centralizer in the whole group Yes