General affine group of degree one
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.
|(field size)||(underlying prime, field characteristic)||Order||Second part of GAP ID|
|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|
Below, is the size of the field and is the underlying prime (the characteristic of the field). We have where is a positive integer.
|order||order of semidirect product is product of orders: The group is a semidirect product of the additive group of , which has order , and the multiplicative group of , which has order (because it comprises all the non-identity elements).|
|exponent||Non-identity elements in the additive group have order and all elements outside it have order .|
|derived length||2||The derived subgroup is the additive group. The exception is the case , where the group is abelian and has derived length 1.|
|Fitting length||2||The Fitting subgroup is the additive group of the field, and the quotient is an abelian group.|
|Frattini length||1||For , we can find two maximal subgroups of order with trivial intersection. Note that this also follows from it being a Frobenius group.|
Arithmetic functions of a counting nature
|number of conjugacy classes||identity element, one conjugacy class of non-identity elements of additive group, conjugacy classes, one for each non-identity coset of the additive group.|
|Property||Satisfied?||Explanation||Corollary properties satisfied/dissatisfied|
|Frobenius group||Yes||The additive subgroup is a Frobenius kernel and the multiplicative subgroup is a Frobenius complement. (note: the case is an exception, where it fails to be a Frobenius group on account of the multiplicative group being trivial).|
|Camina group||Yes||The derived subgroup is the additive group, and every coset of that forms a conjugacy class.|
|abelian group||No||Except the case, where we get cyclic group:Z2|
|nilpotent group||No||Except the case, where we get cyclic group:Z2|
|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 is a prime number rather than a strict prime power.|