#group-theory

[xeter-group]

Exceptional Automorphisms of S_6

The symmetric group S_6 has a special property that S_n does not have for ANY n ≠6. Really? 6, of all numbers?? How odd.

For any group G, and every g in G, the conjugation map f_g: h ↦ g h g^-1 is an automorphism of G. That is, it is an isomorphism from G -> G. The group of these particular automorphisms (under composition) is called the group of inner automorphisms of a group. Did you know that for every symmetric group S_n EXCEPT n=6, this is the entire automorphism group?

For some reason, S_6 has basically ONE (and only one) weird automorphism. What I mean by this is if we denote the inner automorphisms of S_6 by Inn(S_6), then S_6/Inn(S_6) ~= Z_2. (In other words, you can pick a single automorphism sigma such that the automorphisms of S_6 that are not inner automorphisms can be written as a composition of inner automorphisms and this one sigma.)

So we have this weird situation where |Aut(S_2)|=|Aut(C_2)| = 1, which is kinda trivial, |Aut(S_6)| = 2n!, and |Aut(S_n)| = n! otherwise. Kinda weird, huh?

What is also interesting is the proofs that there are no outer automorphisms for n≠6. Basically you can eliminate all but finitely many n, then you can pick off cases, and you are left with an n=6 shaped hole in your cases that CANNOT be filled, which feels so weird to me. If I was proving this myself I'd be going crazy having proven it for every case except 6, and having to resort to some proof which for some reason doesn't work for finitely many values.

Construction

There are so many different ways to construct the weird automorphism of S_6 - I have some links at the bottom. I particularly like the graph theory/geometric ones. Something about using factorisations of K_6 or a dodecahedron just makes the 6 feel more unique. I will admit I don't think I understand what is fundamentally special about 6 enough yet, though, on a philosophical level.

Practically, though, the constructions all basically boil down to the fact you can put a copy of S_5 inside S_6 in a way that isn't the obvious way, which you can only do for n=6. That is the special part.

After that, you have these 6 cosets of S_5 inside S_6 that S_6 acts upon. In other words, each element of S_6 permutes the 6 cosets of S_5 living inside it. But the group of permutations of 6 cosets is S_6. So we have a mapping from elements of S_6 to S_6 - an automorphism! Is this automorphism an inner automorphism? Each construction shows why they are outer differently, but a common theme is to show that the mapping S_6 to S_6 does not take transpositions to transpositions, but inner automorphisms preserve the cycle type.

Some proofs and examples

The wikipedia article. I like the construction about graph partitions. It does not, however, have much detail sadly.

Fortunately, the graph partitions thing from wikipedia is explained here. It's very short and to the point, also quite nice to look at:

Fairly elementary explanations, followed by more intense ones:

Requires only basic group theory to understand the first few explanations they provide (although it isn't trivial if you just learned group theory). Bonus points for "MyStIc PeNtAgOnS" (capitalisation mine):

Allegedly useful, but I can't grab a copy: (I think its on mathscinet under MR1240362 as per David Leep's personal website's publications section)

Combinatorial Structure of the automorphism group of S6 by T.Y. Lam and David B. Leep, Expositiones Mathematicae11 (1993), no. 4, pp. 289-903.

The comment by Matthew Towers here is also interesting:

[chloesimaginationthings]

The FNAF 4 bullies have had a rough few years

[knotty-et-al]

Visualization of the Rubik's cube

[muzetrigger]

*sniff* alright… second post about Nevermore tonight:

We got the group shot! And doesn’t it parallel season one nicely?

Ep. 114 vs. Ep. 39

If season one was about Lenore pulling her group together and Annabel’s backstabbing each other, I think season two might be the reverse.

We’ve already seen Ada, Annabel, and Prospero go up to bat for each other and Lenore’s group has just split up!

[janmisali]

now if you thought "group" is an unhelpful and nondescriptive name for a mathematical concept. just remember that the broader category of mathematical concepts that groups belong to is called, a "category". so it could always be worse

[delicate-sketch]

Idk if this is unpopular opinion or not but while Percy is like “fuck this demigod stuff I am retired” I think Nico would be working in the underworld in the future like I just have this vision of him in a mix of modern and ancient greek clothing doing missions for Hades collecting some troublesome souls idk helping with the judgement talking shit with the dead making the playlist for their elevators vibing to tragedy by bee gees he kinda enjoys the demigod stuff even if its hella dangerous or whatever

[huggywuggysuppy]

love Gem so much. mentioning onstream the past few weeks (hers and other peoples) how there's not a lot of expectation to upload during February... been editing Impossible MC... oh she'll see you all in March...

And posting on the final day of February. absolute power move.

[ringosmistress]

[albino-parakeet]

”When you and the bestie are both obsessed with Dark Web Diving”

These dinosaur chains I wanted to buy for my boots sold out and I’m forever mad at myself for not buying them earlier.

So I drew Ben and Brooklynn with them on their boots, to make me feel better. (You can’t really see it on Ben cause of shading rip.)

Timelapse undercut:

Songs used Blister in the Sun Cover by Crummy Stuff (literally what I thought Ben was singing to in that one episode) and Deceptacon by Le Tigre.

[justpizzaisgood]

Someone once pointed out the similarities between these two promo arts, and i think that it could contain have some really good symbolism.

They could represent Pomni's and Ragatha's relationships with the other cast members as the show progresses.

Let's start with Pomni. When we're introduced to her, we see her as an outsider to the circus. She's lost, confused, and scared. But as the show progresses, we watch her forming deeper bonds with the rest of the cast, starting with Ragatha, then Kinger, then Gangle, and finally Jax. Jax is the most significant representation of this because we see him as an antagonist at the show's start from how he pushes around and bullies Pomni when she was still adjusting. Yet from the latest episode, it's kind of flipped and she seems to be the only one he trusts enough to have deepish conversations with.

Ragatha on the other hand, is introduced as sort of the mom of the group. She tries to comfort Pomni and gives her a tour. She helps her adjust to the adventures and tries to stop Jax from antagonising her and Gangle. She partipates in group activities a lot: the adventures, Pomni's breath-holding experiment, softball practice with Gangle and Jax. Based on all this, we think, she must be the most established person in the cast right? She should have some connection with the rest, since she acts like the leader/mom of the group.

Unfortunately, the opposite is slowly being revealed. First, Zooble hints that she's too nice to criticize Caine's adventures. Then Gangle admits that Ragatha doesn't always seem genuine. Then Jax reveals that her being too nice and happy makes him feel like she's trying to take advantage of him. We slowly realize that she isn't as connected to the others as she first seems. It turns out, she doesnt seem to have deep connections with anyone in the group, save for Pomni and maybe Kinger. It also feels like Ragatha sees herself as slowly losing connections with everyone. The end of episode 5 visually stresses this:

How does this tie in with the promo art?

Let's assume Pomni is watching the sun RISE. Sunrises represent a new day, with new opportunities, and therefore change. Even thoughshe seems to start the day alone, she has the chance to change that. And it shows. As the day (the show) progresses, Pomni makes friends and is adapting to the wackiness of the circus.

As for Ragatha, she's sitting alone at night. Nighttime can be seen as a time of rest, contemplation, and even as a moment to spend time with loved ones or friends after a busy day. Yet Ragatha spends it alone. Even the moon isnt present in Ragatha's promo art, while it seems like the sun is greeting Pomni. Both in the episode's end and in the promo, Ragatha seems to be contemplating the reality of her relationships with others. At the end of the day, Ragatha is all alone (literally in episode 5 lol), even though the during the day it doesn't seem like so.

So yeah that's what i got from it: the promo art represents the two's developping relationships with the cast members, and how they're going in opposite directions.

Could be just a theory and the two promo arts are just promo art. But who knows.

[a-very-tired-jew]

The irony of making that post about stochastic terrorism and how the far Left and Right are both using it is that I had/have representatives of both political extremes coming at me in comments and my inbox. It's not exactly horseshoe theory because one was trying to get me to say that anyone who uses the phrase "eat the rich" is committing stochastic, which is not the case. It was very clear they were sealioning for an answer they wanted and when they didn't get it they did the usual thing and became accusatory and went on the offensive.

The others, because there have been multiple, are doing exactly what I've laid out previously that pro-Hamas antisemites do, which is defend support of terrorist groups as if it's a First Amendment right. Which is only a defense to a certain point. Once you fall under the material support section of the law (e.g. distributing materials produced and approved of by the terrorist organization, actively spreading their rhetoric in conjunction and coordination with them, and inciting and promoting violence according to said rhetoric) then your speech and actions are no longer protected.

So many people want to either believe that they know more about First Amendments rights than they actually do or purposefully misinterpret what stochastic terrorism means so they can label any and all criticism against their blorbo. In the former situation we have laws regarding hate speech and terrorism that address said actions and rhetoric (US 18, 2339A and 2339B for the latter). In the other situation it’s a classic defensive argument that rejects nuance for an absolutist view that rejects any self reflection.

I can work with the former as most people are actually ignorant as to what extent FA covers, but the latter is just arguing from a place of ignorance and digging your heels in like a stubborn child.

[the-sslimmest-shady]

THIS IS A MESSAGE TO THE MALEVOLENT FANDOM: WHY DOES YOUR GUY LOOK LIKE JON FROM THE MAGNUS ARCHIVES. I CANT KEEP MALEVOLENT OFF MY DASH BECAUSE I KEEP ACCIDENTALLY INTERACTING WITH MALEVOLENT FANART THINKING ITS JONATHAN SIMS. I DONT KNOW WHAT MALEVOLENT IS. I KNOW NOTHING ABOUT IT. I DONT WANT TO. THE FANART IVE SEEN IS BEAUTIFUL SO PROPS TO YOU FOR THAT BUT IM JUST VERY CONFUSED. IS THERE SOME KIND OF FANDOM OVERLAP?? IDK HIS NAME BUT HE LOOKS EXACTLY LIKE JON. SLIGHTLY DARK SKINNED DUDE WITH MESSY MID-LENGTH HAIR AND A LOOK OF FEAR IN HIS EYES. YOU KNOW THE ONE. WHO IS HE

[pokemon1oadvanced]

Crazy that season 1 of camp cretaceous brooklynn was mad at sammy from hiding something important from the group which caused tension between the campers to season 3 of chaos theory where sammy is mad at brooklynn for hiding something important from the group causing tension between the campers.

[immortaledd]

Why are we listening to Charlie and not THE GUY WHO LIVED WITH THE VIRUS FOR 40 YEARS??

Please read this in a tone of someone screaming behind a glass window in a straight jacket - not as an authoritative guardian scolding you. I am tweaking out, not crashing out

“Eclipse is acting weirdly because he’s already violent, so it’s turning him into the opposite.”

this isn’t an Opposite Virus, it’s a Ruin Virus, a Virus specifically made to turn people violent to unleash a large kill rate.

The Virus stems from Patient Zero, Ruin.

Ruin- when they were Sun and Moon, is supposed to be a kill-bot to fit The Creator’s goals. The Creator created DCA’s to be murderers. The Virus that stemmed (OUT OF THE CREATOR’S CONTROL) was LIKELY created BECAUSE OF THAT CODING.

Montgomery Gator is ALSO a violent animatronic. He is not CONSTANTLY aggressive or angry like Eclipse is, but he is probably the most violent-by-default we have seen affected by the virus. He is quick to conclusions, revenge, insulting or degrading, but typically to people he believes deserves it. For example, the time Montgomery tried to KILL RUIN when Ruin was “cured” in TSAMS. He only became MORE. VIOLENT.

THAT’S WHY IT’S SO WEIRD!!! THAT’S WHY RUIN’S SO DAMN CONFUSED! THAT’S WHY RUIN HIMSELF IS SCARED!

You can NOT be taking the opinions from CHARLIE.

She is a child, she has no idea what she’s even dealing with, she has never had prior experience to The Virus or even her own MAGIC before, and Charlie has VERY OFTEN created these ideas that were PROVEN to be false. WHEN SHE PROPOSED THE IDEA that “it affects everyone differently”, she was IMMEDIATELY PROVEN WRONG THROUGH HER OWN CONTRADICTIONS!

Stepping away from in-character, from a writer standpoint -

Davis created the Virus narrative, Davis created Ruin’s story and the character himself, Davis knows what’s going to happen and why things are going the way they’re going.

LISTEN TO “THE DIRECTOR”’S CHARACTERS. Whoever is CONTROLLING THE STORY is the actor you pay MOST ATTENTION TO!!!!

[chemsexholmes]

one thing worth noting in the conversation about whether or not college students have "become stupid" is that the attitude of the general public toward academic study has largely shifted in the past few decades away from education as a moral betterment (and also, implicitly, a status symbol), toward totally embracing the idea that education is the Thing that gets you Jobs. which of course is pushed by policymakers and academic institutions, because how else do you sell academic funding in a capitalistic society? but has created an ethos about education as something you have to drag your feet through in order to get your degree, which makes students feel more inclined to use LLM like chatgpt to complete their work because they fundamentally aren't connecting with the material.