K4 Groups And The Octal

A K4 group (actually the "Klein Four-group" or C2 x C2, the product of two cyclic groups of order two) is a very simple structure. It could be considered to be isomorphic to the ordered pair (a,b) where addtion of (a,b) + (c,d) is defined as (a+c, b+d) and the elements {a,b,c,d} take their values from the set {0,1}. Addition is done modulo two so that 1+1 = 0 mod 2.

Then the table of the additive operation is given as

Where the elements a = (0,1), b = (1,0), c = (1,1) are as in the table and the identity element 0 = (0,0). We simply note that aside from use of the identity element, each sum of two different elements is equal to the remaining one of the three. An element added to itself is 0, the identity, and of course 0 + x = x as usual.

The thought may spring up that the values {0,1} in the ordered pair (a,b) as initially defined may be alike to a "truth-table" . However it is not binary "true/false" that is in view here, rather it is more generally the symmetry of "odd and even" where odd is "1" and even "0". One can produce a similar table for a power set of any set of two elements under symmetric difference, replacing zero with the empty set. That does not strictly occur in the bible's revelation text but is only in view when the element forming the empty set (identity) is made analogous to "hurting the oil or the wine".

The octal group or product C2 x C2 x C2, the product of three groups of the cyclic group of order two is similar, we again define it as in component-wise addition done modulo two on the ordered triple (a,b,c) where each of {a,b,c} is also taken from {0,1}where 1+1 = 0 mod 2.

We obtain a similar table thus:

Note the embedded K4 subgroup within the red lines. There are seven such K4 subgroups in the octal group above, formed from the identity element with the triples [a,b,c], [a,d,e], [a,f,g], [b,d,f], [b,e,g], [c,e,f], [c,d,g] under closed addition. (There are also seven order two subgroups formed of zero and each of the seven singletons, [0,a].[0,b],[0,c], etc...)

We would simply refer to such order four subgroups as K4 subgroups.

Lastly with one more observation, we may make an addition on the K4 subgroups of the octal as follows

[0,a,b,c] + [0,a,d,e] = [0,a,f,g]

Where the last remaining group of the product is the only subgroup that has the same common element (which is "a" here).

We have a more formal definition that this operation is the complement of the symmetric difference taken in the octal (A v B)^c.

I.e. [0,a,b,c] + [0,a,d,e] = {0,a,b,c,d,e,f,g} - {b,c,d,e} = [0,a,f,g]

Then, we may use the octal group itself for the identity element rather than zero.

[0,a,b,c,d,e,f,g] + [0,a,d,e] = {0,a,b,c,d,e,f,g} - {b,c,f,g} = [0,a,d,e]

Lastly, we may perform a similar operation on each K4 subgroup using the group itself as the identity in addition upon subgroups of the K4 group isomorphic to C2.

[0,a] + [0,b] = {0,a,b,c} - {a,b} = [0,c]

and [0,a,b,c] + [0,c] = {0,a,b,c} - {a,b} = [0,c] etc.

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