Static Subgroups

To start with we define the frobenius map on a finite field of characteristic two as Frob: x => x^2. This has the effect that every element of those finite fields is sent to another, (except for the identity elements that are sent to themselves.) and is done so in a 1-1 maner that preserves the isomorphism conditions; that frob(x+y) = frob(x) + frob(y) and frob(x*y) = frob(x) * frob(y).

When we use a seven cycle for multiplication, we note that the sets of elements {x,x^2,x^4} and {x^3,x^5,x^6} are permuted amongst themselves. Note that the set {x,x^2,x^4} inside our cycle are (as capitialised) the second, third and fifth elements listed in the cycle "x" when we start with unity thus; x = (a,B,D,c,F,g,e) if a=1 and we note that [b,d,f] is a subgroup. Likewise aside from requiring a specific choice of unity, we may keep the same seven cycle with say d = 1 then (d,c,f,g,e,a,b) requires [c,e,f] as fixed under the frobenius map.

We also note that using the frobenius map that ther remaining elments are also sent to themslves (as capitalised) thus: (a,b,d,C,f,G,E) - and {c,g,e} is also permuted amongst itself. This is similar but instead of "x" we are using the inverse "x^6".

In truth the frobenius map frob: x=>x^2 holds fixed the elements (0,1) which forms the prime subfield. The triples permuted above are not "fixed" in proper terms, but are in this section called "static".

It is possible to move unity from one element to another and the resulting static elements under frobenius divide the cycle into static subgroups. The static subgroup never contains unity, and we may simply define two octals upon which that particular seven cycle preserves the structure of subgroups. For instance if x=(a,b,d,c,f,g,e) = (b,d,c,f,g,e,a) etc.

a=1 => [b,d,f] and {c,e,g} is fixed
b=1 => [c,d,g] and {a,e,f} is fixed
d=1 => [c,e,f] and {a,b,g} is fixed
c=1 => [a,f,g] and {b,d,e} is fixed
f=1 => [b,e,g] and {a,c,d} is fixed
g=1 => [a,d,e] and {b,c,f} is fixed
e=1 => [a,b,c] and {d,f,g} is fixed

We can most certainly form an octal from either the column of subgroups in triples [X,Y,Z] or from those of {X,Y,Z}. In both cases we state that the seven cycle (or as it were under frobenius) holds as static the subgroups [...] and {...} in the octal to which they belong, with the corresponding choice of unity. In this sense we consider the unity element to "float".

Likewise there are four choices of unity that may correspond to a static subgroup (to which unity does not belong) as well as there being four static subgroups with which a particular choice of unity may be associated. There are two instances of each with a separate C7 group for a particular octal, for instance:

if a=1 and [b,d,f] is static then, (listing {b,d,f} in cyclic and anticyclic order) (a,b,d,c,f,g,e) and (a,b,f,c,d,g,e) are generators of two different C7 groups which preserve the structure of subgroups of our octal.

It is important to know that two different C7 groups acting on the same field GF(8) are only isomorphic up to relabelling elements. I.e. there is no frobenius isomorphism between the fields if they have the same symbols for the octal and the same operation of addition. Multiplication gets "changed out" rather than "corrupted" - we may safely assume that these describe the same octal or additive part, but not the same complete finite field under the frobenius map. Of course they are isomorphic using a more general "relabelling" for a map instead.

Continue To Next Page

Return To Section Start