Metamath

2:  Trinity In The Father
Let us continue.It is perhaps easy enough to give the tables for addition and multiplication for our groups belonging to a finite field, so lets do so.
It is perhaps easier and more worthwhile to make the exact representation of the field of order eight as thus: (with the irreducible x^{3}+x+1=0)
With these associations, for brevity we will drop the "0"'s from the listings of subgroups. ther are seven subgrops of order 4, all Klein fourgroups, the sets {a,b,c},{a,d,e},{b,d,f},{b,e,g},{c,d,g},{c,e,f}. There are eight groups of seven cycles that map these groups together coherently. An example that is contingent upon that particular representation of the field above is x^{2} = (a,f,d,g,b,c,e).
mapping between the groups we find these eight sets are:
(a,b,d,c,f,g,e)
(a,b,d,e,g,c,f)
(a,b,e,c,g,f,d)
(a,b,e,d,f,c,g)
(a,b,f,c,d,e,g)
(a,b,f,g,e,c,d)
(a,b,g,c,e,d,f)
(a,b,g,f,d,c,e)
We simply state that for every multiplicative element, and every additive element, there is a Klein four group corresponding to those elements in such a manner that there exists a multiplication and an addition for each of those elements so as to perform the identical operation. We associate additive elements with the Klien four groups under the operation of the complement of the symmetric difference of the sets of elements of those subgroups. Thus {a,b,c} + {a,d,e} = {a,f,g}. etc.
Then for any pair of Klein four groups as subgroups above, there is a unique klein four group that can be added to either to transform on to the other. There is also an element of a group of seven cycles tht will accomplish the same thing.
In fact, we may associate fourgroups of subgroups easily enough by their intersection. We may also find some elements where there is no correspondence. For example we may rearrange the elements as;
There are 168 such arrangements where there is no intersecting correspondence. The choice would seem fairly arbitrary, but for every arrangement where the structure of the correspondence is set, there are eight such isomorphic arrangements where the singletons form a similar group.
Likewise as before, we may pick any one of our groups to be our unity element, and we may just as happly relabel the singletons a through g as we desire and find an isomorphic arrangement. Where we ascribe a klein four group such as {a,b,c} to the groups of subgroups, just as arbitrarily as we may choose to do so. It follows that there must be at least one klein four group that shares a correspondence with a singleton, one of its own elements, eg:
As with {a,b,c}=a above. We may pick our unity but unless we refuse an isomorphic structure, at least one element must be fixed as a member of a subgroup. Happily without much trouble every two subgroups are mapped to each other by just one other: and multiplicatively there is a unique element making the same result. We are in general more interested in correspondences of subgroups to singletons they do not contain. We now move on to the case where we examine seven cycles.
{a,f,g} = {a,b,c,d,e,f,g}  ({a,b,c} v {a,d,e}) = {a,b,c} + {a,d,e}
and multiplicatively there is a unique element making the same result. We now move on to the case where we examine seven cycles.
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