All Questions
Tagged with gr.group-theory lo.logic
22
questions
111
votes
2
answers
15k
views
Does every non-empty set admit a group structure (in ZF)?
It is easy to see that in ZFC, any non-empty set $S$ admits a group structure: for finite $S$ identify $S$ with a cyclic group, and for infinite $S$, the set of finite subsets of $S$ with the binary ...
- 3,481
29
votes
3
answers
3k
views
Is it decidable whether or not a collection of integer matrices generates a free group?
Suppose we have integer matrices $A_1,\ldots,A_n\in\operatorname{GL}(n,\mathbb Z)$. Define $\varphi:F_n\to\operatorname{GL}(n,\mathbb Z)$ by $x_i\mapsto A_i$. Is there an algorithm to decide whether ...
- 18.1k
22
votes
2
answers
2k
views
What is the largest Laver table which has been computed?
Richard Laver proved that there is a unique binary operation $*$ on $\{1,\ldots,2^n\}$ which satisfies $$a*1 \equiv a+1 \mod 2^n$$
$$a* (b* c) = (a* b) * (a * c).$$
This is the $n$th Laver table $(A_n,...
- 3,607
38
votes
5
answers
4k
views
A “mother of all groups”? What kind of structures have "mother of all"s?
For ordered fields, we have a “mother of all ordered fields”, the surreal numbers $\mathbf{No}$, a proper-class “field” which includes (an isomorphic copy of) every other ordered field as a subfield. ...
- 1,657
14
votes
1
answer
882
views
Amenability and ultrafilters
Among hundreds of equivalent definitions of amenability (for discrete, countable, groups), I would like to discuss two which are most common:
A1. A group $G$ is amenable if it admits a Folner ...
- 30.8k
2
votes
4
answers
531
views
relatively free groups in $Var(S_3)$
Suppose $S_3$ is the symmetric group of order 6. Which elements of the variety $Var(S_3)$ are relatively free?
This question is related to my previous question
Relatively free algebras in a variety ...
- 2,183
81
votes
3
answers
5k
views
How do I verify the Coq proof of Feit-Thompson?
I probably don't have the appropriate background to even ask this question. I know next to nothing about formal or computer-aided proof, and very little even about group theory. And this question is ...
- 28.8k
38
votes
4
answers
2k
views
On sentences true in all finite groups
Let $w$ be a group word with two variables $x$ and $y$.
Is the sentence $(\forall x)(\exists y)w=1$
true in every group if it is true
in every finite group?
The same question about the sentence $(\...
- 893
33
votes
1
answer
1k
views
Is this conjecture strictly weaker than P=NP?
My three computability questions are related to the following group theory question (first asked by Bridson in 1996):
For which real $\alpha\ge 2$ the function $n^\alpha$ is equivalent to the Dehn ...
user6976
22
votes
6
answers
2k
views
Is any interesting question about a group G decidable from a presentation of G?
We say that a group G is in the class Fq if there is a CW-complex which is a BG (that is, which has fundamental group G and contractible universal cover) and which has finite q-skeleton. Thus F0 ...
- 1,189
21
votes
4
answers
1k
views
Is there a non-Hopfian lacunary hyperbolic group?
The question's in the title and is easily stated, but let me try to give some details and explain why I'm interested. First, a disclaimer: if the answer's not already somewhere in the literature then ...
- 23.6k
17
votes
2
answers
2k
views
Why are model theorists so fond of definable groups?
My PhD was on so called "pure" model theory, and my advisor was not very much interested in applications of model theory to algebra. Now I feel the need to fill in the gap, and I'd like to educate ...
- 935
15
votes
2
answers
897
views
Element being trivial in a finitely presented group independent of ZFC
Is there an explicit finitely presented group $G$ and an element $g\in G$ such that the statement "$g$ is equal to the identity" is independent of ZFC?
user178109
13
votes
0
answers
379
views
On sentences true in all finite groups (revisited)
Let $w$ be a group word with variables $\bar x, \bar y$, where
$\bar x=(x_1,\dots ,x_m)$ and
$\bar y=(y_1,\dots ,y_n).$
I am interested in the following questions.
(1) Is the sentence $(\forall\bar ...
- 893
12
votes
2
answers
539
views
Do there exist acyclic simple groups of arbitrarily large cardinality?
Recall that a group $G$ is acyclic if its group homology vanishes: $H_\ast(G; \mathbb Z) = 0$. Equivalently, $G$ is acyclic iff the space $BG$ is acyclic, i.e. $\tilde H_\ast(BG;\mathbb Z) = 0$.
In ...
- 59k
9
votes
3
answers
1k
views
First-order axiomatization of free groups
Is there a way to axiomatize [non-abelian] free groups in first-order logic using the language of groups (which contains the binary operation symbol $\cdot$, and the constant symbol $e$)?
Is there ...
- 37.7k
9
votes
1
answer
739
views
What classes of groups can arise as "symmetry groups of terms"?
Let $\mathfrak{A}$ be an algebra (in the sense of universal algebra). To each term $t(x_1,...,x_n)$ in the language of $\mathfrak{A}$ in which each variable actually appears we can assign a group $G_\...
- 19k
6
votes
1
answer
219
views
Are $G$-limits of a slender group $G$ in the space of marked groups also slender?
A group $G$ is slender if every subgroup $H \leq G$ is finitely generated. This includes polycyclic-by-finite groups. Such groups are also called noetherian.
Suppose that $L$ is a $G$-limit group in ...
- 963
5
votes
1
answer
494
views
Translating first order statements about symmetric groups into the language of numbers and back
A question I was asked recently lead me to the following question. For every closed first order formula $\theta$ in the group signature consider the set $N_\theta$ of natural numbers $n$ such that the ...
user6976
3
votes
1
answer
332
views
Is a finitely generated residually free group "almost LERF"?
Let $G$ be a finitely generated residually free group.
(i.e. for each $1 \neq g \in G$ there exists a homomorphism $\tau \colon G \to F$ such that $F$ is a free group, and $\tau(g) \neq 1$.)
Let $...
- 11.2k
2
votes
0
answers
70
views
Empty preimage under homomorphism of finitely presented groups with decidable word problems
Let $G, H$ be finitely presented groups with decidable word problems.
Can there be a homomorphism $f:G\to H$ such that there is no algorithm deciding given $w\in H$ whether $f^{-1}(w)$ is empty or not?...
user178109
2
votes
1
answer
220
views
Possible symmetry groups of power terms
Previously asked and bountied at MSE:
Let $\mathfrak{E}=(\mathbb{N};\mathit{exp})$ be the algebra in the sense of universal algebra consisting of the natural numbers with just exponentiation. To each ...
- 19k