New answers tagged big-list
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Statements in differential geometry independent from ZFC
An explicit construction from the undecidability problem is this. There is a computer program $C$ that can check whether a (completely formal) proof in ZFC is valid, i.e. whether each step is either ...
15
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Statements in differential geometry independent from ZFC
[Using the comments for context on undecidability/independence of ZFC]
A computably undecidable problem is whether or not a homology sphere has a metric of positive scalar curvature [Page 79 of ...
6
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Popular mistakes in probability
The most popular mistake made by students is assuming that a sum of uniform random variables is uniform.
A related mistake, done at a more advanced level, is assuming that independence is needed to ...
2
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Results from abstract algebra which look wrong (but are true)
A theorem of Bass: For a ring $R$, every left $R$-module has a projective cover if and only if $R$ satisfies the descending chain condition on principal right ideals.
1
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Results from abstract algebra which look wrong (but are true)
A subring of a Noetherian ring need not be Noetherian.
Given all the stability properties that Noetherian rings enjoy, this may sound surprising at first, but it becomes much more obvious if you think ...
4
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Results from abstract algebra which look wrong (but are true)
If $k$ is an algebraic number field then for every positive integer $n$ there exist infinitely many field
extensions of $k$ of degree $n$ having no proper subfields over $k$.
10
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Results from abstract algebra which look wrong (but are true)
There exists a finitely-generated infinite group with only two conjugacy classes, a difficult result of Osin.
17
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Results from abstract algebra which look wrong (but are true)
As fields, the algebraic closures of the fields ${\bf Q}_p$ are isomorphic, and are isomorphic to the complex numbers.
5
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Results from abstract algebra which look wrong (but are true)
There is some theory about maximal valuation rings (a special type of ring with linearly ordered ideals, not necessarily a domain) and then there are almost-maximal valuation rings which is of course ...
17
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Results from abstract algebra which look wrong (but are true)
For a group with finitely many elements of finite order, the set of elements of finite order is a subgroup.
2
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Results from abstract algebra which look wrong (but are true)
In the same vein of the statement the OP included:$$G\times H \cong G\times K \Longrightarrow H\cong K$$ for product of finite groups, which can be rephrased as "product in finite groups is ...
3
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Results from abstract algebra which look wrong (but are true)
That there exist finitely presentable non-Hopfian groups. [I still remember my shock when I was first learned this result!]
A group G is Hopfian if every surjective homomorphism $\phi:G\to G$ is in ...
13
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Results from abstract algebra which look wrong (but are true)
In combinatorial group theory, loosely speaking almost any problem one can imagine, in full generality, turns out to be undecidable. This includes the word problem, the isomorphism problem, the ...
6
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Results from abstract algebra which look wrong (but are true)
Quite a few things in the Hopf algebra world are surprising:
Takeuchi's theorem: Every connected graded bialgebra is a Hopf algebra. (No finiteness assumptions!) Takeuchi was actually more general: ...
9
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Results from abstract algebra which look wrong (but are true)
It seems to me appropriate to name the following totally unexpected result, which is too good to be true, yet is true.
Are there only finitely many finite groups with $m$ generators of
exponent $n$, ...
6
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Results from abstract algebra which look wrong (but are true)
I suppose there is a case for saying that Jordan's theorem on finite complex linear groups might be such a result: there is a function $f: \mathbb{N} \to \mathbb{N}$ such that for every $n \in \mathbb{...
34
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Results from abstract algebra which look wrong (but are true)
Let $G$ be a finite group and $n \mid |G|$.
If $S = \{x \in G : x^n = 1\}$ contains exactly $n$ elements, then $S$ is a subgroup of $G$.
There seems no a priori reason to expect $S$ to be a subgroup ...
7
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Results from abstract algebra which look wrong (but are true)
The Auslander–Buchsbaum theorem that every regular local ring is a unique factorization domain.
I should say that the first time I saw this theorem stated, I was not immediately surprised, but that ...
2
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Results from abstract algebra which look wrong (but are true)
A simple module $S$ over a finite dimensional algebra $A$ over an algebraically closed field $K$ such that there exists a non-split short exact sequence $0\rightarrow S \rightarrow X \rightarrow S \...
61
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Results from abstract algebra which look wrong (but are true)
The free group with infinitely many generators is a subgroup of the free group with two generators.
21
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Results from abstract algebra which look wrong (but are true)
Every finite index subgroup of a finitely generated profinite group is open. The converse is obvious, but this direction was quite surprising to me. This is a result of Nikolov and Segal and uses the ...
9
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Results from abstract algebra which look wrong (but are true)
In a finite Frobenius group, the set of all fixed point free elements together with the identity forms a subgroup.
This might not have such a shocking effect to us, since usually when we first hear ...
3
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Results from abstract algebra which look wrong (but are true)
An example might be that the category of abelian groups is hereditary. That is, every complex of abelian groups is quasi-isomorphic to the (graded) direct sum of its cohomologies. Although this is ...
23
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Results from abstract algebra which look wrong (but are true)
The Nielsen-Schreier theorem that subgroups of free groups are free might have seemed surprising from am algebraic view given the analogue for many other algebraic structures is false. While this is ...
34
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Results from abstract algebra which look wrong (but are true)
Every finite simple group can be generated by at most $2$ elements. This is another famous consequence of the classification.
23
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Results from abstract algebra which look wrong (but are true)
Every element of a finite simple non-abelian group is a commutator. This is the positive solution to the Ore conjecture (see Liebeck, O’Brien, Shalev, and Tiep - The Ore conjecture) and uses the ...
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Examples of errors in computational combinatorics results
It is well known that the Appel–Haken–Koch proof of the four-color theorem was controversial because of its use of an electronic computer, but it is not as well known that the original proof had many ...
2
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Places where one can post open problems
For open problems in group theory: https://arxiv.org/abs/1401.0300
0
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Examples of errors in computational combinatorics results
A regular map of type $\{ 3, 6 \}$ is one for which every vertex has degree $6$ and every face has degree $3$. Define $\chi(v)$ to be the number of regular maps (up to isomorphism) of type $\{ 3, 6 \}$...
3
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Examples of common false beliefs in mathematics
“A reversible computer can factor integers efficiently in polynomial-time”: Since a reversible computer can multiply two integers efficiently in polynomial-time, it can also factor an integer ...
11
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Examples of errors in computational combinatorics results
About 20 years ago, the number of groups of order 1024 was reported to be 49487365422 in "A millennium project: constructing small groups", and this number was repeated in other sources. ...
3
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Places where one can post open problems
The website https://www.scilag.net/ is also meant as a database.
3
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Places where one can post open problems
The Open Problems Project (TOPP) is
focussed on discrete and computational geometry.
We (Erik Demaine, Joe Mitchell, and I) started it in 2001 but its $78$ problems are now only sporadically updated ...
3
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Examples of errors in computational combinatorics results
The Ramsey number survey by Radziszowski (Small Ramsey Numbers, revision 16 ) has a couple of footnotes mentioning incorrect values. Unfortunately there is not much information on the cause of or ...
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Examples of errors in computational combinatorics results
The thread Widely accepted mathematical results that were later shown to be wrong? contains combinatorial examples, for example
the Perko pair in knot enumeration [Perko, Kenneth A. jun., On the ...
6
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Examples of errors in computational combinatorics results
The Catalan numbers have a famous generalization associated to finite irreducible reflection groups. Afaik, the formula
$$\operatorname{Catalan}(W)=\prod_{i=1}^n \frac{d_i+h}{d_i}$$
appeared first in ...
3
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Collecting proofs that finite multiplicative subgroups of fields are cyclic
Here is a proof that $\mathbb F_q^\times$ is cyclic. Let the finite extension $F/\mathbb Q_p$ have residue field $\mathbb F_q$. The group $\mu_{q-1}=\{x\in F:x^{q-1}=1\}$ is isomorphic to $\mathbb F_q^...
8
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Examples of errors in computational combinatorics results
(1) Number of n-dimensional Bravais lattices: correct sequence here https://oeis.org/A256413, incorrect sequence here https://oeis.org/A004030
From the paper Opgenorth, J; Plesken, W; Schulz, T, ...
7
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Places where one can post open problems
If it's a problem in Number Theory, the annual West Coast Number Theory meetings have a problem session, and the problems get collected & edited & posted to https://westcoastnumbertheory.org/...
17
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Examples of errors in computational combinatorics results
(1) In this paper (published J. Combinatorial Designs, 15 (2007) 98-119), in the history section starting page 3, we cite many published errors in counting Latin squares and related objects. Some, but ...
6
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Places where one can post open problems
Arnold Mathematical Journal has a problems section.
9
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Places where one can post open problems
I am one of the moderators of the Open Problems in Algebraic Combinatorics blog.
First of all, we welcome submissions from anyone who has a good open problem in algebraic combinatorics that they want ...
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Accepted
Places where one can post open problems
If you can motivate the problem and make some partial progress on it, you can try and publish it as a paper in a specialized journal, or at the very least upload it to the arXiv.
If you only have ...
8
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Places where one can post open problems
If you have half-a-dozen open problems in some definite area, why not write an article where you explain them, why there are important, what are the difficulties ? I did that once (Five open problems ...
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Places where one can post open problems
Open Problem Garden
I occasionally stumble across this in my search results. It's currently skewed heavily to graph theory, which suggests that the user base also skews that way. Con: it doesn't ...
3
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Unnecessary uses of the axiom of choice
The highly-upvoted, accepted answer (by Theo Johnson-Freyd) to another MO question, Why worry about the axiom of choice?, points out that the usual proof of the Poincaré–Birkhoff–Witt theorem assumes ...
7
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What are some interesting applications/corollaries of Kleene's Recursion theorem?
Here is a further example, which I find to be one of the rather more philosophically profound results in computability theory. Namely, Rice's theorem.
The theorem states, at bottom, that no nontrivial ...
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8
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What are some interesting applications/corollaries of Kleene's Recursion theorem?
I like Wehner's version of the Slaman-Wehner construction.
Theorem. There is a family of sets which can be enumerated by every non-computable oracle, but is not computably enumerable.
The family is $\...
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What are some interesting applications/corollaries of Kleene's Recursion theorem?
Here is another of my favorite uses of the Kleene recursion theorem.
It arises from Turing's remarkable 1936 paper, "On computable numbers...", in which he defines Turing machines, provides ...
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19
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What are some interesting applications/corollaries of Kleene's Recursion theorem?
My favorite use of the Kleene recursion theorem is the universal algorithm.
In the baby form, consider the program $e$ that (on any input) undertakes the following process: it looks for a proof from ...
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