Symmetry groups are beautiful and fascinating.

Cayley diagram for the Symmetry Group of a Cube, 

We can write some simple equations based on "following arrows" in the diagram. If red is "a" and blue is "b", we can say many things. "i" is the identity. If we take a sequence of arrows starting from a point and if the sequence of arrows lands us back at the starting point, then the path forms a closed loop and is equivalent to not having left the point at all.

a^4 = i
(ab)^3 = i
(abb)^2 = i
(aabb)^2 = i

Note that "a" and "b" are equivalent. They can be interchanged in the equations. Hence we may also conclude

b^4 = i
(ba)^3 = i
(baa)^2 = i
(bbaa)^2 = i

Similarly any "a" can be replaced by "-a", the reverse path, or any "b" by "-b". Also not that "aaa" is the same as "-a". Hence we can write:

aaa = -a

And hence,
(aaab)^3 = ((-a)b)^3 = i
(aaabb)^2 = ((-a)bb)^2 = i

Any sequence that forms a loop (equals i) can be started at any point (ie can be rotated) and it still forms a loop. Hence the following are equivalent.

(aba)^4 = i = aba.aba.aba.aba = aab.aab.aab.aab = (aab)^4.

hence we have,
(aab)^4 = i
(baa)^4 = i
(abba)^2 = i

Of course, I have missed some. You can fill them in with what you know now.

Now if we select "c" to be some string a and b, such that c^3 = 1, for example c = ab, we can draw a new cayley diagram from the resulting triangles and one of the squares (example a^4 = 1). The "c" arrows are drawn in green here.

This maybe visualized as a solid. A cube with each vertex replaced by a triangle and each edge having the width of the triangle's side. Now a very good diagram of the front view:

The above group, corresponding to the diagrams, has a name - it is the symmetry group S4. It corresponds to the permutations of 4 character tuple, (A,B,C,D). How many permutations are there? There are 24. These can be though of as the 24 different points on the above diagram.

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