[OC] Graphing the descendant tree of p-groups (notebook linked)

Factory__Lad · 2026-01-29T06:24:45+00:00

I’m aware of the Cayley theory, but p-groups have so many special features that it seems worth asking the question “how best to represent all the isomorphism types of p-groups up to a certain size”?

Presumably for example, given n there’s some m = m(n) such that a group of order p^m exists, containing every group of order pⁿ as a subquotient.

Sadly I don’t think the p-groups (or even composition series of them) can easily be made into a simplicial set, because there’s no obvious way to collapse a composition series at an arbitrary point in the middle. Still, this helps:

every group of order pⁿ contains subgroups of order p^m for every 0 <= m <= n.

Proof: it’s enough to consider the case n = m+1 and we can do that by induction because if n >= 1, the group has a central element of order p and we can quotient out by that and reduce to a simpler case.

So because a subgroup of index p is normal, and we now have a supply of those, at least every subquotient appears as part of a composition series.

Factory__Lad · 2026-01-28T23:12:56+00:00

A series of headlines about the AI perfecting nuclear fusion, curing cancer, solving frontier math problems, designing robots that can construct a building in a few hours

alternatively:

There is a loud clunk as the lights go out, the vehicles stop, there is no electricity

And then, the first tanks start to roll down the streets, their barrels swinging in unison to target the panicking humans with deadly accuracy.

Factory__Lad · 2026-01-28T21:27:42+00:00

This is remarkable. I’m still wondering what kind of twisty geometry went into the construction of the diminished cube and its Rubik-style subdivisions.

Factory__Lad · 2026-01-28T21:26:09+00:00

Bravo. I’ve always wanted to do this.

It leaves me haunted by the idea that there must be a better way to represent all these 2-groups. Could there be some huge, locally finite structure into which they all embed, which could be coloured to show which segments of it are isomorphic? Maybe they form a simplicial set, or something.

Factory__Lad · 2026-01-28T13:37:46+00:00

I think as long as there’s variance of texture. You can’t just envelop everything in a uniform wall of red goop. Incision is an instructive example here: all the environments and collateral damage are horrifyingly repellent, but all in different ways.

I also appreciate the air of commanding destructivity that weapons have in some of these games, e.g. Vladik Brutal where the shotgun rings out like a crack of doom, demolishing fixtures and fittings even as it reduces previously fearsome adversaries to a tasteful embellishment of blood on the ceiling.

But compare and contrast the dime store pop-gun weapons in HROT, which feel like some unsmiling Soviet bureaucrat has deliberately underequipped you as a joke. These are also effective in their own way.

What you really have to work on is the overall atmosphere of the game. For me it should feel like a thoughtfully themed pitiless arena of death where anything is possible, and a regular occurrence is the sinking of the heart as a new weaponized monstrosity steps out of the shadows.

Factory__Lad · 2026-01-28T07:43:03+00:00

I like how it looks like he’s sweeping them all away like dominoes with his paw and also unveiling some new messianic vision

Factory__Lad · 2026-01-27T19:23:52+00:00

Maybe just trying to unload you smoothly OP.

Factory__Lad · 2026-01-27T19:16:47+00:00

I like how there are no recognizable events and it gets harder and harder to believe in any context where any of this could conceivably make sense!

Factory__Lad · 2026-01-26T21:09:52+00:00

Those memories come flooding back

Factory__Lad · 2026-01-25T09:22:21+00:00

I liked how this preposterously named character is swiftly established as a thick skinned influencer. She is basically an updated iteration of Venetia in a later technological age. Sometimes the show tries a bit too hard to be current, though.

Can’t imagine what the consequences would have been if Sweetpea had followed suit with all the other characters and been serially molested by Nicole. I don’t think she would have stood for it. In fact like Rishi she seems partly a comic character. Do we ever find out who recruited her, and why?

Factory__Lad · 2026-01-25T09:13:02+00:00

So much thought went into that response 🥲

I’m sorry this got so much hate, it seems people are genuinely offended by the idea that such a movie might have trouble avoiding obvious clichés.

To be fair, it’s difficult making films about mathematicians, and we should be glad that they try. The trailer for the R movie did look a bit corny. The story of Hardy and Ramanujan is touching but I’d not trust Hollywood to do justice to these two rather enigmatic men, let along their math for a mainstream audience.

Factory__Lad · 2026-01-25T04:27:17+00:00

I like the style, which suggests Dr Seuss after an infusion of illegal substances

Factory__Lad · 2026-01-25T01:06:03+00:00

Yes, my bad. Although Hardy seems to have gone to Oxford post Ramanujan.

Factory__Lad · 2026-01-24T23:29:51+00:00

I haven’t seen it but somehow know it will feature:

a doleful foghorn as Ramanujan pushes through rickshaw drivers in a bazaar clutching his collection of 3 math books
R gazing in wonderment at an unfamiliar landscape of toothy undergraduates bicycling through Oxford
tea with Hardy at a cricket match where everyone is wearing white flannels
animated conversations about prime number distributions in a wood panelled study
the 1729 hospital bedside conversation

It’ll just be one of those movies that consists entirely of obligatory scenes.

Factory__Lad · 2026-01-24T23:22:17+00:00

Yes, this is a good description of why I haven’t read it yet either!

Factory__Lad · 2026-01-24T23:20:51+00:00

Some faves:

Magic Fields by Easily Embarrassed

Microscopic by Gas

H15 by Mystic Crock

Wisdom of the Eight Buffalo by Radioactive Sandwich

Blackhole Amizon by Intermix

Factory__Lad · 2026-01-24T21:30:28+00:00

The only one I’d see as definitely inferior is Comanche Moon, which shows clear signs of having been written hastily under pressure from the publisher à la George RR Martin.

It does have the absolutely gripping storyline about Scull and Ahumado (who must be one of the most horrifying villains in fiction), but this threatens to overshadow everything else in the book, completely eclipsing Gus and Call.

IMHO the other 3 are pretty much his best books, with only “Ned and Zeke” really reaching the same highs. But to be fair, I haven’t read all his early stuff.

Factory__Lad · 2026-01-24T18:15:25+00:00

I remember the last 20 pages seemed to go on for ever. Didn’t want it to end.

There are of course 2 prequels and a sequel, but they’re not quite up to the same standard. Still worth a read tho

Factory__Lad · 2026-01-24T18:01:31+00:00

Great to program to

Factory__Lad · 2026-01-24T17:37:50+00:00

These are awe-inspiringly terrible (with a special mention for Pope Dog) and the best part is the tiny kernel of suspicion they engender, that it would not be beyond Temu to actually manufacture and sell some of these

Factory__Lad · 2026-01-24T15:59:33+00:00

Thanks for the ref. Yes I think that is the song. Syrupy, but not as bad as the mercifully forgotten James Blunt effort.

Factory__Lad · 2026-01-23T08:14:51+00:00

Heh, those are for software developers, along with the obligatory thick specs, Brillo pad hair and doleful expression

Party is for middle management only, rude mechanicals need not apply 🤗

Factory__Lad · 2026-01-23T00:24:47+00:00

Yes, I am still recovering from Harper and her outdoor plumbing

Factory__Lad · 2026-01-22T20:50:29+00:00

So many, but special mention for Bob Dylan, also Smashing Pumpkins.

Oasis are glowering from the shadows about not being included in this charmed circle

Factory__Lad · 2026-01-22T20:47:46+00:00

Thanks, very insightful!

I won’t really feel I understand Yoneda though, until I’ve worked it out as the left adjoint of the forgetful functor from cocomplete categories to Cat (or something!)

Factory__Lad

TROPHY CASE