Element structure of general semilinear group of degree one over a finite field
This article gives specific information, namely, element structure, about a family of groups, namely: general semilinear group of degree one.
View element structure of group families | View other specific information about general semilinear group of degree one
This article describes the element structure of the general semilinear group of degree one. Recall that, for a field , the group is defined as:
where is the multiplicative group of
,
is the prime subfield of
, and
denotes the Galois group of
over
.
We are interested specifically in the case where is a finite field of size
, in which case the group is written as
. Suppose
is a prime power with underlying prime
, so that
for a positive integer
.
is the characteristic of
. In this case,
is cyclic of order
(see multiplicative group of a finite field is cyclic) and
is cyclic of order
(generated by the Frobenius map
).
Thus, is a metacyclic group of order
with presentation:
(here denotes the identity element).
Note that if , the group is the same as
and is cyclic of order
.
Summary
Item | Value |
---|---|
order | ![]() |
number of conjugacy classes | ![]() ![]() |
Number of conjugacy classes formulas
For every fixed value of , the number of conjugacy classes simply becomes a polynomial in
. The values of these polynomials for small
are listed below:
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Polynomial in ![]() |
---|---|
1 | ![]() |
2 | ![]() |
3 | ![]() |
Particular cases
![]() |
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general semilinear group ![]() |
order of the group (= ![]() |
number of conjugacy classes (polynomial in ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
element structure page |
---|---|---|---|---|---|---|
2 | 2 | 1 | trivial group | 1 | 1 | -- |
3 | 3 | 1 | cyclic group:Z2 | 2 | 2 | element structure of cyclic group:Z2 |
4 | 2 | 2 | symmetric group:S3 | 6 | 3 | element structure of symmetric group:S3 |
5 | 5 | 1 | cyclic group:Z4 | 4 | 4 | element structure of cyclic group:Z4 |
7 | 7 | 1 | cyclic group:Z6 | 6 | 6 | element structure of cyclic group:Z6 |
8 | 2 | 3 | general semilinear group:GammaL(1,8) | 21 | 5 | element structure of general semilinear group:GammaL(1,8) |
9 | 3 | 2 | semidihedral group:SD16 | 16 | 7 | element structure of semidihedral group:SD16 |
Conjugacy class structure
For generic r
This table is complicated! We consider conjugacy classes in the multiplicative group and conjugacy classes outside the multiplicative group. Note that .
Nature of conjugacy class | Number of rows of this type | Size of conjugacy class for each row | Number of such conjugacy classes for each row | Total number of elements for each row | Total number of conjugacy classes | Total number of elements |
---|---|---|---|---|---|---|
in the multiplicative group and in the prime subfield | 1 | 1 | ![]() |
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(one such row for every positive divisor ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() |
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(one such row for every positive divisor ![]() ![]() ![]() ![]() ![]() ![]() |
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Total | -- | -- | -- | -- | ![]() |
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For r = 2
We take :
Nature of conjugacy class | Size of conjugacy class | Number of such conjugacy classes | Total number of elements |
---|---|---|---|
in the multiplicative group and in the prime subfield | 1 | ![]() |
![]() |
outside the prime subfield | 2 | ![]() |
![]() |
outside the multiplicative group | ![]() |
![]() |
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Total | -- | ![]() |
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For r = 3
We take :
Nature of conjugacy class | Size of conjugacy class | Number of such conjugacy classes | Total number of elements |
---|---|---|---|
in the multiplicative group and in the prime subfield | 1 | ![]() |
![]() |
outside the prime subfield | 3 | ![]() |
![]() |
outside the multiplicative group | ![]() |
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Total | -- | ![]() |
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