Ciro Santilli @cirosantilli 35

It sees and moves individual atoms!!!

Once Ciro was at a University course practical session, and a graduate was around helping out. Ciro asked if what the graduate did anything specifically related to the course, and they replied they didn't. And they added that:Even though Ciro was already completely disillusioned by then, that still made an impression on him. Something is really wrong with this shit.

Other people that think that the educational system is currently bullshit as of 2020:

By Ciro Santilli: Section "How to teach".

Others:

As per en.wikipedia.org/w/index.php?title=Semidirect_product&oldid=1040813965#Properties, unlike the Direct product, the semidirect product of two goups is neither unique, nor does it always exist, and there is no known algorithmic way way to tell if one exists or not.

This is because reaching the "output" of the semidirect produt of two groups requires extra non-obvious information that might not exist. This is because the semi-direct product is based on the product of group subsets. So you start with two small and completely independent groups, and it is not obvious how to join them up, i.e. how to define the group operation of the product group that is compatible with that of the two smaller input groups. Contrast this with the Direct product, where the composition is simple: just use the group operation of each group on either side.

Product of group subsets

So in other words, it is not a function like the Direct product. The semidiret product is therefore more like a property of three groups.

The semidirect product is more general than the direct product of groups when thinking about the group extension problem, because with the direct product of groups, both subgroups of the larger group are necessarily also normal (trivial projection group homomorphism on either side), while for the semidirect product, only one of them does.

Conversely, en.wikipedia.org/w/index.php?title=Semidirect_product&oldid=1040813965 explains that if $G=N⋊H$, and besides the implied requirement that N is normal, H is also normal, then $G=N\times H$.

Smallest example: $D_6=C_3⋊C_2$ where $D$ is a dihedral group and $C$ are cyclic groups. $C_3$ (the rotation) is a normal subgroup of $D_6$, but $C_2$ (the flip) is not.

Note that with the Direct product instead we get $C_6$ and not $D_6$, i.e. $C_3\times C_2=C_6$ as per the direct product of two cyclic groups of coprime order is another cyclic group.

TODO:

Bibliography: math.stackexchange.com/questions/1726939/is-this-intuition-for-the-semidirect-product-of-groups-correct

This is a general philosophy that Ciro Santilli, and likely others, observes over and over.

Basically, continuity, or higher order conditions like differentiability seem to impose greater constraints on problems, which make them more solvable.

Some good examples of that:

How are the bands measured experimentally?

Why are there gaps? Why aren't bands infinite? What determines the width of gaps?

Bibliography:

Inward Bound by Abraham Pais (1988) page 282 shows how this can be generalized from the Maxwell-Boltzmann distribution

Lots of demos.

Easily classified as the direct product of cyclic groups of prime order.