Our notation: $D_n$, called "dihedral group of degree n", means the dihedral group of the regular polygon with $n$ sides, and therefore has order $2n$ (all rotations + flips), called the "dihedral group of order 2n".

The reason public relations is evil in modern society is because, like discrimination, public relations works by dumb association and not logic or fairness.

If you're the son of the killer, you're fucked.

This is unlike our ideal for law which attempts, though sometimes fails, at isolating cause and effect.

As shown in Video "Simple Groups - Abstract Algebra by Socratica (2018)", this can be split up into two steps:This split is sometimes called the "Jordan-Hölder program" in reference to the authors of the jordan-Holder Theorem.

Good lists to start playing with:

History: math.stackexchange.com/questions/1587387/historical-notes-on-the-jordan-h%C3%B6lder-program

It is generally believed that no such classification is possible in general beyond the simple groups.

Bibliography:

TODO why do we care about this?

Note that if a group is k-transitive, then it is also k-1-transitive.

TODO this would give a better motivation for the Mathieu group

Higher transitivity: mathoverflow.net/questions/5993/highly-transitive-groups-without-assuming-the-classification-of-finite-simple-g

The Royal Society's Nobel Prize.

royalsociety.org/grants-schemes-awards/awards/copley-medal/ says it is now open to international citizens, but having a quick look at the 2010 awards still suggests that it is very British centric, or at least anglophone centric, much like the society fellowship itself. That's likely the reason why the Nobel prize won, being much more international from the start.

Good background: www.theguardian.com/science/the-h-word/2016/oct/07/prize-science-medal-history-nobel

Minimum number of elements in a generating set of a group.

The Nobel Prize of computer software or hardware!

More like a "lifetime achievement" though, rather than the Nobel Prize, which tends to be for more specific achievements.

How to build it: math.stackexchange.com/questions/3137319/how-in-general-does-one-construct-a-cycle-graph-for-a-group/3162746#3162746 good answer with ASCII art. You basically just pick each element, and repeatedly apply it, and remove any path that has a longer version.

Immediately gives the generating set of a group by looking at elements adjacent to the origin, and more generally the order of each element.

TODO uniqueness: can two different groups have the same cycle graph? It does not seem to tell us how every element interact with every other element, only with itself. This is in contrast with the Cayley graph, which more accurately describes group structure (but does not give the order of elements as directly), so feels like it won't be unique.

Take the element and apply it to itself. Then again. And so on.

In the case of a finite group, you have to eventually reach the identity element again sooner or later, giving you the order of an element of a group.

The continuous analogue for the cycle of a group are the one parameter subgroups. In the continuous case, you sometimes reach identity again and to around infinitely many times (which always happens in the finite case), but sometimes you don't.

If your kids are about to starve, fine, do it.

But otherwise, Ciro Santilli will not, ever, spend his time drilling programmer competition problems to join a company, life is too short for that.

Life is too short for that. Companies must either notice that you can make amazing open source software projects or contributions, and hire you for that, or they must fuck off.

Companies must either notice that you can make amazing projects or contributions, and hire you for that, or they must fuck off.

Two ways to see it:

A ring where multiplication is commutative and there is always an inverse.

A field can be seen as an Abelian group that has two group operations defined on it: addition and multiplication.

And then, besides each of the two operations obeying the group axioms individually, and they are compatible between themselves according to the distributive property.

Basically the nicest, least restrictive, 2-operation type of algebra.

Examples: