Metamath

An Application
Without stating in depth, the "Four Colour Theorem" holds for a 2D plane dividied into regions, so that when colouring the regions only four colours are required so that no two neighboring regions meeting with an edge share the same colour.
If we label each region as one of white, red, black or green (w,r,b,g) then we may denote the colour changes across an edge as (wr,wb,wg,rb,rg,bg)
We can also likewise use the group (an octal) {0, w+r, w+b, w+g, r+b, r+g, b+g, w+r+b+g}
setting unity equal to w+r+b+g, we can find the associations between subgroups as follows
If; X = (g+b, r+b, r+b, 0)
Y = (g+b, g+w, b+w, 0)
W = (r+b, r+w, b+w, 0)
U = (r+g, r+w, g+w, 0)
Then in analogy under a seven cycle (w+r+b+g)=>(r+b)=>(r+g)=>(w+g)=>(b+g)=>(w+r)=>(w+b)=>(w+r+b+g)
(w+r+b+g) = 1 (unity) which corresponds to X, say.
r+b = U
r+g = Y
w+g = W+U
b+g = W
w+r = Y+W
w+b = Y+U
which is our resultant octal reversal.
likewise, unity (w+r+b+g) could correspond to either X,Y, W or Z
The octal under frobenius then has the following property.
If X = unity say, then holding all white regions fixed we cycle the three remaining colours. There are two ways the maths works, as there are two possible C7 groups that allow this.
likewise there are two seven cycles holding Y fixed. These correspond to holding all red regions fixed and cycling the three remaining colours.
The action of the eight different C7 groups that hold fixed each of the four groups X,Y,W and U (two each) when (w+r+b+g) = 1 are essentially isomorphic; yet differ so little in their pairs the cycling of the colours is equivalently true.
Under subgroup addition, the other three groups (W+U etc) form a Klein four group with the octal as identity as well as C3 under the action of frobenius. The equation X+Y = W+U obviously holds under Klein four group addition on subgroups.
(we assumed all along that r+b+r+b = 0 (or r+r=0) holds for w,r,b,g.)
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