Metamath

Factoring to A5
We know that the group A5 is the set of all even elements of S5, and there is also a coset of odd elements, each of these have 60 possible permutations from the total 120 (By the theorem of Lagrange).
We can consider the logical ability to solve the problem of finding our door keys as simply (see,touch) since it is a usual part of our experience, there is nothing keeping us from performing the action of reaching through the window if they are indoors. However if "touch" is somehow impossible  and our desire is to use logic to get around this problem, ie there may be great heat, we either cease action or we get burned. Our desire may be for (see,touch) to be familiar as it would be in normal situations, however we can reduce the problem to one of two sets, A5 and its coset.
We may have all the elements of A5 as well as another coset (of all elements of A5 in product with say, (sense,lust)). the A5 set will include (see,touch)(see,smell) for example. Because the set A5 contains even elements, we are appearing to find another way around the problem, without employing logic but using conjugation on A5. Almost as if smelling the chocolate bar would help us find it when it is out of sight.
For the most part we will ignore the coset of A5, but we will note that the inverse cycle of (a,b,c,d,e) is (e,d,c,b,a). And because A5 is a normal subgroup of S5 if H is the group A5 and g and its inverse g^{1} are elements of S5 then gHg^{1} is also an element of H = A5 for any element g. (g^{1} is uniquely defined for g.),
So the process of choosing H or A5 then pre and post multiplying by g and g^{1} is like (see,touch,smell,taste) H (see,taste smell,touch) for any H in A5. Where (see,touch,smell,taste) is in S5. Thus we can with the appearance of considering the desire of our flesh, the (see,touch,smell,taste) part apply our logic to seeing if there is any other way around the problem (our elements in H).
Lets simplify this. consider (see,touch) H (see,touch) The right most element is the "lust part" where one would desire to overcome an antagonism. The left most element is the inverse, (actually self inverse for a transposition) The elements in H are mapped on to some other elements in H = A5 here by this conjugation, but all products in this conjugation are in A5..
Thus in considering the problem without the need for logic to say "leave well alone" when there is an antagonism, considering both sides of "see then touch" or "desire touch following seeing" There appears to be sensual justification in there always being another element of A5 for solving the problem.
I can justify this simply, as confusion between there being an actual sensual ability to sin, over the thought that "I saw then touched quite naturally" because "I desired to touch it since i saw it, and I couldn't see a sensual reason not to." The reduction to A5 has the appearance of disregarding "as many elements as possible" all belonging to the coset of A5 containing our antagonism (sense,lust), while still maintaining the sensual desire to disobey (our elements g and g^{1}). The conjugate element yet within A5 is part of reasoning upon the senses, and not standing in the commandment. It is the senses last stand in giving an appearance of reasoning whether the senses (with the pride of life) have an environmental or sensual stand against obedience.
Likewise obedience is not found in the coset of A5. The coset has a multiplication of each individual element (on one side) from A5 by the transposition being considered. Concentrating on the subgroup A5 is the natural thing to do when avoiding the temptation. In the beast system we have the appearance of examining the system to see whether there is "room for uptake" on the leavening influence. However, there are elements (even numbers of transpositions remember) that contain the transposition we factor down by, So it is really true that a little leaven leavens the whole lump. We can call this llittle leaven in A5 "the lust of the flesh".
However, this system is inherently sinful, and reasoning upon the senses against a commandment brings you away from the absolute of the fixed position of obedience. The system here upon A5 is method to sin, and if the law should be considered in this manner, we consider breaking it. Paul wrote, "What shall we say then? Is the law sin? God forbid. Nay, I had not known sin, but by the law: for I had not known lust, except the law had said, Thou shalt not covet."
So where would logic step in to put a stop to this sensual ability to disregard consequence aka "thou shalt not covet"?
An inverse of a cycle is simply the reversed order product of its transpositions, and so we can order a series of "I saw then touched, I saw then smelled, I saw then tasted" on the left as with the "lust part" of "I tasted since I had seen, I smelled since I had seen, I touched since I had seen" on the right.
The critical part of the argument is (see,touch) may be forbidden as sin, "thou shalt not eat of it, neither shalt thou touch it or look upon it".
There is a linkage in the cycle that roots the desire to familiarly taste what is seen, and quite logically if there is a good reason not to, a commandment, then one end of the cycle is interrupted as a fixed element that may not be permuted. Logic in itself would and should step in to say " there is no way it can be eaten".
If on the other hand the question of whether the sensual desire outweighs our consideration of the commandment, and the logic is put into the effort to examine only the senses, then if logic becomes simply the factoring down to A5, to find simply what is desirable to the senses as the divining rod of truth, the outcome is sin.
So if we consider the full group S6 with 720 elements, 120 of those elements are purely the elements of S5 comprising permutations of the senses only. (sending logic to itself.) In dropping this group to A5 by conjugation we now have in our construction 660 elements, where either the problem solving logic is free to act without an antagonism, or to consider the transposition of an antagonism (sense,lust) as whether to break a commandment. The first 600 elements which all include a logical aspect are the "Hath God not said ye shall eat of every tree in the garden?" as the serpent said, whereas the 60 lost elements are the minds attempt to filter out the disobedience whilst the mind searches for sensual freedom to disobey and eat the forbidden fruit. (with the lust of the flesh.)
The commandment is purely physical, so when the logical part is not included in this sensual A5, we say that these 660 elements are the effort to compare that one commandment with the freedom we have in other situations,.. As we might be commanded not to eat of a tree but would be free to eat of any other tree as Eve relayed to the serpent. This is the "lust of the eyes" part. The breaking of the commandment does not care for the logic employed in reasoning from the senses, this A5 does not include permutations of the logical sort of the other 600. Logic, stands with the commandment here, but freedom in other situations is employed in the temptation process. The factoring down is then only upon the senses, (since the commandment was the antagonism on the purely physical act) and not upon the freedom or ability to solve a problem in other situations.
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