2-Sylow subgroup of general linear group:GL(2,3)
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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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Contents
Definition
is the general linear group of degree two over field:F3. In other words, it is the group of invertible
matrices with entries over the field of three elements. The field has elements
with
.
is one of the 2-Sylow subgroups of
, i.e.:
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 | ![]() |
center of general linear group:GL(2,3) | cyclic group:Z2 |
normal core | ![]() |
2-core of general linear group:GL(2,3) | quaternion group |
normal closure | the whole group | -- | general linear group:GL(2,3) |
characteristic core | ![]() |
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 | ![]() ![]() ![]() ![]() ![]() |
The subgroups have order ![]() |
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) |
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 |