Seven Spirits Of God

Carrying over from the Metamath section we hold the multiplicative group of the finite field of eight elements as like to the Holy Spirit.

Essentially in seven symbols a to g we can cycle the elements with "x" = a->b->d->c->f->g->e with e->a.

Then the particular octal group that has the subgroups {(a,b,c),(a,d,e),(a,f,g),(b,d,f),(b,e,g),(c,d,g),(c,e,f)} (where any one of the triples is the product of its two companions.) has its subgroups preserved by that cycle above which we simply write (a,b,d,c,f,g,e).

We see that now we have

(a,b,c)=>(b,d,f)=>(c,d,g)=>(c,e,f)=>(a,f,g)=>(b,e,g)=>(a,d,e)=>(a,b,c) (again) as a cycle of the subgroups.

In the Galois field with eight elements GF(8) there is the frobenius automorphism that holds the subfield (0,1) fixed, but there is also a subgroup which is sent to itself which should be called "static" but we will also call "fixed" (even though its elements are cycled). It is not possible for this subgroup to contain the unity element in GF(8) since the automorphisms form a "Galois" group of order three; that is; were the unity element to be in the fixed subgroup, then the two companions of unity would have to swap positions (like an order 2 symmetry) in order to hold static the subgroup but 2 does not divide 3 (the order of the Galois group), so the static or fixed subgroup never contains unity.

In fact there are four subgroups that would not contain unity, (whichever element is chosen to be unity.) We can arbitrarily choose an element as unity, say "a". Then we may consequently hold static each of the subgroups {(b,e,g),(b,d,f),(c,e,f),(c,d,g)} by simply constructing our elements in the right way or by simply relabeling them.

Simply put, no matter the choice of unity, within the cycle used, the fixed subgroup is comprised of the first, second and fourth element in the cycle after unity. So therefore (a,b,d,c,f,g,e) with a=1 would hold fixed the subgroup (b,d,f), whereas if c=1 then (a,f,g) is static.

In multiplicative terms, if x is our seven cycle then the automorphisms are formed by the elements x, "x squared" and "x to the power of four". Each of them are calculated as follows.

x = (a,b,d,c,f,g,e)
x^2 = (a,d,f,e,b,c,g)
x^4 = (a,f,b,g,d,e,c)

Where ^ is used for "to the power of". Note that x^8 = x, so repeated squaring of the multiplicative cycle renders the static subgroup to go from (b,d,f) => (d,f,b)=>(f,b,d).

Knowing that the elements in the fixed (static) group are formed by the first, second and fourth elements after unity in the written cycle we can construct a total of 48 seven cycles that may act on an octal group to preserve the cycling of the octal's subgroups.

First, pick a unity element, ("a" here) Then form two sets for each potentially static subgroup as follows (a,B,D,_,F,_,_) and (a,B,F,_,D,_,_). Then shift unity from the first element to the second and onward, writing in the subgroups in order to preserve them as follows:

(a,b,d,C,f,_,_) then (b,d,C,f,G,_,a,) with b unity, then (d,C,f,G,E,a,b) with d unity - then we have made (a,b,d,c,f,g,e). Likewise we can reorder the static subgroup and find (a,b,f,c,d,e,g).

So we have eight possible groups isomorphic to C7 that preserve the subgroups of each octal group. We show a generator of each here.

(a,b,d,c,f,g,e) with (b,d,f) fixed
(a,b,f,c,d,e,g) with (b,d,f) fixed
(a,c,d,b,g,f,e) with (c,d,g) fixed
(a,c,g,b,d,e,f) with (c,d,g) fixed
(a,c,e,b,f,g,d) with (c,e,f) fixed
(a,c,f,b,e,d,g) with (c,e,f) fixed
(a,b,e,c,g,f,d) with (b,e,g) fixed
(a,b,g,c,e,d,f) with (b,e,g) fixed

The 48 seven cycles are generated by the six non-identity powers of these eight elements. They form eight separate groups isomorphic to C7. (Each of the eight share the identity element which sends every element to itself.)

Each of the symbols 'a' through 'g' represent one spirit of God. Each C7 group "represents a church" or a lampstand. We shall see how the angel or messenger that goes from church to church represents a mystery that has this beautiful structure traced together. (The mystery is that there are only seven churches!)

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