Why Serpents, Why Not Just Scorpions?
There is a reason surely why the text has two groups of spiritual forces acting against the spirit of grace in Jesus Christ. Why not for example have the one force that imposes S5 upon the "refining" churches of the foolish virgins, and then turn every operation inward? The operation of the octal underneath the C7 group is the spiritual reality underlying the same elements in C7. The locusts that are given "five months" in which to act are given the same five elements in the octal, the additive group. These elements or "months" are fixed and immobile, and the enemy seeks to seduce away believers from the octal group and the group C7.
In the building of the image to the beast the false prophet does indeed set up the system A5, but we must note that it is synthetic, a combination of both serpents and scorpions. There is no operation of an extension field over GF(8) that can provide a set of additive subgroups under a multiplicative orbit which has a "sub-orbit" of cycle length 2. That is, we can not use an extension field to provide a subgroup of order 2.
The multiplicative group of a characteristic 2 finite field has an odd order, so can not be divisible by 2.
Then if we have a set of subgroups in a cycle of order 2, we would surely have
G = xH
H = xG
that is,
G = Gx^2
So even if we have a short orbit then the set {1, x^2, x^4, x^6, x^8 ... etc} fixes that particular orbit. Since we have a "shortened" orbit we must have a factor the order of a subfield. I.e. G or H is the additive group of a subfield of the parent extension F. Then we note that even if the sequence {1, x^2, x^4, x^6, x^8 ... etc} is itself such a subfield, it has the factor group in F* of {1,x} or |F*| is divisible by 2. Of Course |F*| is odd, and this is a contradiction.
The serpents provide the ability to form a transposition, by opposing the octal underneath with what is in effect the original ultrafilter under a transposition of two of its elements. (Thus negating the need for a change in multiplicative terms, this being "additive").
What is more in point is whether a field of any prime power order could have a subgroup in F* of order char(F). Then use of the frobenius map would send every element of the subgroup of order p=char(F) to unity, which is most improper - every homomorphism of fields must be either trivial or an isomorphism.
Of course, what of the concept of algebraic closure? It is not apparent that there could not be a short orbit of length 2, if the infinite sequence of squared terms {1, x^2, x^4, x^6, x^8 ... etc} can form a short orbit. An infinite multiplicative subgroup could act just as well, and there would be a "smallest subfield" that would do the job were it not for the comment on subgroups of order char(F) above.
We would require then that the infinite series {1, x^2, x^4, x^6, x^8 ... etc}form a subfield, but that {1, x} not be a subgroup of F*. A seeming impossibility, and indeed it is so. (It being a factor group of a "free group" - as a cofactor of a subgroup of another free group.)
So the destroyer, "Abaddon" or "Apollyon", the angel that rises from the bottomless pit surely goes to its own destruction. It can not fully frustrate the spirit of God found in GF(8). The additive transpositions raised by the serpents are shown in the text (by the passage of the two witnesses) that such witness of God can stand firm and be in the right, until such time as the serpents contradict themselves. There can be no agreeing to disagree with the enemy, but the "fact" of the reality of Jesus witnessed to by the two men in the text is clearly not an inconsistency.
The ingredients for the factoring down to A5 are present in full in the serpents and scorpions combined, but they can not coexist in the same manner or spiritual form - they are two opposing spirits that collide and "ordo ab chao" the result of factoring down results within the system of the false prophet set up in the centre. The Laodicean church that rises up on its beast from the pit is in direct worship to the knowledge of evil, the "mystery of iniquity".
We could just use another characteristic field, but if we allow these other characteristics, then we would immediately see short orbits of the field's subfields. The locusts are given that they should torment men five months - therefore we could consider just fields with a subfield of index 5 in its parent field's multiplicative group.
However, the number of additive subgroups of a given order of such a field is divergent and increases very rapidly. There is no "closed" sequence as there are in GF(2)+, GF(4)+ and GF(8)+. We would have a very chaotic result - but not in frustration of the operations of GF(8) - The intent of the enemy is to seduce worshippers away from God, not to confuse them entirely.
In using other characteristics a subgroup of an extension should therefore have an appropriate short orbit - again we require a subfield of orbit length 2, and a field over that subfield must be of index 2. Then we require p=2 otherwise the short orbit is not possible, The orbit is always too long: If we were to use a "large" additive group that were also index 2 in its orbits, we again require a subfield, this time in the cofactor. The factor group of the sequence {1, x^2, x^4, x^6, x^8 ... etc} is still needed.
We can in other characteristics find short orbits acted upon by even order multiplicative subgroups of F, but not of the whole field upon its subfields: The index of F+ over our G is too large to restrict ourselves to an orbit of length 2, (because p>2) when we MUST use the whole field. We may as well take the order two element in F* and restrict ourselves to that: but in creating our simple short orbit with this square root of unity, we have restricted ourselves to operating with an additive group only, not using the full multiplicative group.
In a short orbit we include every element of F in one group position (more than one for long orbits) or in that of its "cofactor" - but of length 2? We have a multiplicative subgroup also of "cycle length 2", but there is no additive subgroup of index 2 in F+. This is the limit to which we can restrict ourselves - all elements of F must be in either our H or G: so acting from any one to any other in either group would open up the use of the whole field: It would be nonsense to have "null - multiplicative" elements mixed in with our even order subgroup's elements.
We have reduced ourselves to an equivalent of splitting the form of the field of odd characteristic or then to using the serpents: So why bother with these other characteristics? That is as close to a reason for the serpents as there is: We would most certainly require characteristic 2 for the correct index of subgroups. Then we have no other possible "chariots", they need to be separate or disjoint (except for 0) - as they are short orbits. Then there is the need for serpents in order to construct transpositions. Then the serpents exist because they are simpler to construct, and just as iniquitous.
The locusts hurt no green thing neither any tree, that is they do not destroy the "singletons to triples" associations that represent the "trees" alluded to earlier, using other characteristics would surely do these structures "hurt".
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