Draw a 2 peak “symmetric” function in powerpoint. E.g. the outer line, above. Copy it, but scale it down, overlaying it on the former. E.g. the second outer line above. Repeat a few times, noting the “oscillations” that appear as you add the functions.
Now skew the picture (using non-commutative operators). Pick the right side of the innermost rectangle (with all selected), and drag it leftwards. Note the result. The oscillations on ONE SIDE line up!
When thinking about what is… an alternating group, within a symmetric group, thinking binary really helps.
The outer curve is 00000 on the left, and 11111 on the right. The next inner is 0000 (left), 1111 (right). Of course, this eventually comes down to 0 (left) and 1(right) in which the peaks represent the minimum quantum of change. Or, the alternating “core” – or parity – of a symmetric space.
If one uses walsh functions instead of symmetric functions (certain overlayable, orthogonal bit strings of 1s and 0, not just 0000, 1111 etc), now you can do the same thing, if one fiddles with ”characteristic functions,” orthogonal vectors rotating around an equi-probable contour (ellipse) and the associated fourier transform ideas in a discrete group. This gives one superposition of states that – none the less – is still computable.
its not hard to get to RSA (hint) : rotate (normalizer), Symmetric, Alternating.