Ciro Santilli @cirosantilli 40

en.wikipedia.org/w/index.php?title=Mathieu_group&oldid=1034060469#Multiply_transitive_groups is a nice characterization of 4 of the Mathieu groups.

Apparently only Mathieu group $M_{12}$ and Mathieu group $M_{24}$.

www.maths.qmul.ac.uk/~pjc/pps/pps9.pdf mentions:Hmm, is that 54, or more likely 5 and 4?

scite.ai/reports/4-homogeneous-groups-EAKY21 quotes link.springer.com/article/10.1007%2FBF01111290 which suggests that is is also another one of the Mathieu groups, math.stackexchange.com/questions/698327/classification-of-triply-transitive-finite-groups#comment7650505_3721840 and en.wikipedia.org/wiki/Mathieu_group_M12 mentions $M_{12}$.

So simple!! You can either:

A handle cancels out a Möbius strip, so adding one of each does not lead to a new object.

You can glue a Mobius strip into a single hole in dimension larger than 3! And it gives you a Klein bottle!

Intuitively speaking, they can be sees as the smooth surfaces in N-dimensional space (called an embedding), such that deforming them is allowed. 4-dimensions is enough to embed cover all the cases: 3 is not enough because of the Klein bottle and family.

There's exactly one field per prime power, so all we need to specify a field is give its order, notated e.g. as $GF(n)$.

Every element of a finite field satisfies $x^{order}=x$.

It is interesting to compare this result philosophically with the classification of finite groups: fields are more constrained as they have to have two operations, and this leads to a much simpler classification!

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:

$M_n({\mathbb{F}}_q)$ accounts for them all, which we know how to do due to the classification of finite fields.

So we see that the classification is quite simple, much like the classification of finite fields, and in strict opposition to the classification of finite simple groups (not to mention the 2023 lack of classification for non simple finite groups!)

Ciro Santilli is very fond of this result: the beauty of mathematics.

How can so much complexity come out from so few rules?

How can the proof be so long (thousands of papers)?? Surprise!!

And to top if all off, the awesomely named monster group could have a relationship with string theory via the monstrous moonshine?

all science is either physics or stamp collecting comes to mind.

The classification contains:

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 3D regular convex polyhedrons are super famous, have the name: Platonic solid, and have been known since antiquity. In particular, there are only 5 of them.

The counts per dimension are:

The cool thing is that the 3 that exist in 5+ dimensions are all of one of the three families:Then, the 2 3D missing ones have 4D analogues and the sixth one in 4D does not have a 3D analogue: the 24-cell. Yes, this is the kind of irregular stuff Ciro Santilli lives for.

One major application of this classification is that different boundary conditions are suitable for different types of partial differential equations as explained at: which boundary conditions lead to existence and uniqueness of a second order PDE.

Bibliography:

This gate set alone is not a set of universal quantum gates.

Notably, circuits containing those gates alone can be fully simulated by classical computers according to the Gottesman-Knill theorem, so there's no way they could be universal.

This means that if we add any number of Clifford gates to a quantum circuit, we haven't really increased the complexity of the algorithm, which can be useful as a transformational device.

A popular set of universal quantum gates derived from Clifford gates is Clifford plus T.

Set of quantum logic gate composed of the Clifford gates plus the Toffoli gate. It forms a set of universal quantum gates.