General affine group of degree one
Contents
Definition
For a field
For a field , the general affine group of degree one over
is defined as the general affine group over
of degree one. Equivalently, it is the external semidirect product of the additive group of
by the multiplicative group of
, where the latter acts naturally on the former by field multiplication. Explicitly, it is denoted
or
, and can be written as:
Alternative definition as automorphisms of a polynomial ring
For a field , the general affine group of degree one
can be defined as the group
.
Note that this definition does not extend to general affine groups of higher degree. For ,
naturally sits as a subgroup inside
but is not the whole automorphism group.
For a finite number
Let be a prime number and
be a power of
. The general affine group or collineation group
is defined as follows. Let
denote the field with
elements. Then
is the semidirect product of the additive group of
with its multiplicative group.
Equivalently it is the general affine group of degree over the field of
elements.
Particular cases
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Order | Second part of GAP ID |
---|---|---|---|---|
2 | 2 | cyclic group:Z2 | 2 | 1 |
3 | 3 | symmetric group:S3 | 6 | 1 |
4 | 2 | alternating group:A4 | 12 | 3 |
5 | 5 | general affine group:GA(1,5) | 20 | 3 |
7 | 7 | general affine group:GA(1,7) | 42 | 1 |
8 | 2 | general affine group:GA(1,8) | 56 | 11 |
9 | 3 | general affine group:GA(1,9) | 72 | 39 |
Arithmetic functions
Below, is the size of the field and
is the underlying prime (the characteristic of the field). We have
where
is a positive integer.
Function | Value | Explanation |
---|---|---|
order | ![]() |
order of semidirect product is product of orders: The group is a semidirect product of the additive group of ![]() ![]() ![]() ![]() |
exponent | ![]() |
Non-identity elements in the additive group have order ![]() ![]() |
derived length | 2 | The derived subgroup is the additive group. The exception is the case ![]() |
Fitting length | 2 | The Fitting subgroup is the additive group of the field, and the quotient is an abelian group. |
Frattini length | 1 | For ![]() ![]() |
Arithmetic functions of a counting nature
Function | Value | Explanation |
---|---|---|
number of conjugacy classes | ![]() |
identity element, one conjugacy class of non-identity elements of additive group, ![]() |
Group properties
Property | Satisfied? | Explanation | Corollary properties satisfied/dissatisfied |
---|---|---|---|
Frobenius group | Yes | The additive subgroup ![]() ![]() |
|
Camina group | Yes | The derived subgroup is the additive group, and every coset of that forms a conjugacy class. | |
abelian group | No | Except the ![]() |
|
nilpotent group | No | Except the ![]() |
|
metabelian group | Yes | The derived subgroup is the additive group of the field (when ![]() |
Satisfies: solvable group |
supersolvable group | Sometimes | The group is supersolvable if and only if the field is a prime field, i.e., if and only if ![]() |