Some theorems

Posting some theorems, and occasionally other stuff.
By @bahran_cihan

[Some theorems @CihanPostsThms]

NON-THEOREM ANNOUNCEMENT: As someone struggling with depression/isolation, one reason I opened this account is to get some sense of belonging with the math community. I would appreciate if you can refrain from "that is obvious duh", "you miss the point" type of comments. Thanks.

[Some theorems @CihanPostsThms]

@geoffreyirving It seems that the only known examples of this phenomenon have p=5 and P≅C₅.

[Some theorems @CihanPostsThms]

[Conder 2017]. There is a finite group G with a Sylow 5-subgroup P and g∈G–P such that every element of the coset Pg has order 5.

[Some theorems @CihanPostsThms]

A finitely presented residually finite group has a decidable word problem.

[Some theorems @CihanPostsThms]

The AI-bro "Plan accordingly" discourse is insufferable.

[Some theorems @CihanPostsThms]

Given a finite group G, write P(G) for the commuting probability of G, that is, the ratio of commuting ordered pairs in G x G.

[Browning 2023]: Under the reverse ordering of rational numbers in [0,1], the set
{P(G) : G finite}
is well-ordered with order type ω^ω.

[Some theorems @CihanPostsThms]

@selfattentive I don't know you.

To me it is interesting that 42 being a sum of three perfect cubes was only a conjecture until 2019, when the numbers in the OP were found to work.

[Some theorems @CihanPostsThms]

42 = (-80538738812075974)³ + (80435758145817515)³ + (12602123297335631)³

[Some theorems @CihanPostsThms]

[Kirby–Siebenmann]. TOP/PL = K(ℤ/2, 3).

[Some theorems @CihanPostsThms]

42 = (-80538738812075974)³ + (80435758145817515)³ + (12602123297335631)³

[Some theorems @CihanPostsThms]

[Antonyan–Dobrowolski 2015]. For a locally compact Hausdorff group G with a closed subgroup H, TFAE:
(1) G/H is locally contractible.
(2) G/H is a manifold.

[Some theorems @CihanPostsThms]

[Bosboom–Demaine–Hesterberg–Lynch–Waingarten 2015]. It is PSPACE-hard to decide whether Player 1 has a forced win in a two-player Mario Kart race from given starting positions for the players.

[Some theorems @CihanPostsThms]

Every position of Rubik's cube can be solved in at most 20 half-turn moves. Moreover there are positions for which 19 moves is not enough.

[Some theorems @CihanPostsThms]

It is conjectured that all zeros are simple.

[Some theorems @CihanPostsThms]

[Bui–Heath-Brown 2013]. If RH holds, then at least 19/27 of the zeros of the Riemann zeta-function are simple.

[Some theorems @CihanPostsThms]

The upper bound is improved to 46 here:
https://t.co/WWynt3XBfS

[Some theorems @CihanPostsThms]

[Davis 1955]. Given a lattice (L, ≤), TFAE:
(1) Every monotone f:L→L has a fixed point.
(2) L is complete.

[Some theorems @CihanPostsThms]

The product rule follows from the multivariate chain rule.

[Some theorems @CihanPostsThms]

The unique pair consists of the right triangle with sides of lengths (377, 135, 352) and the isosceles triangle with sides of lengths (366, 366, 132).

[Some theorems @CihanPostsThms]

A triangle is called rational if it has rational side lengths.

[Hirakawa–Matsumura 2019]. Up to similitude, there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area.