4
votes
1answer
584 views

solvable word problem without algorithm

Let $G$ be a finitely generated group. I wonder if there are examples where: 1) The word problem is known to be solvable in $G$ but there is no algorithm known. 2) The word problem is known to be ...
48
votes
3answers
2k 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 ...
7
votes
2answers
454 views

What is the name of this type of groups?

Suppose $A$ is a finite set and $\Sigma=A\cup A^{-1}$. Let $L\subseteq \Sigma^{\ast}$ be a regular language on the alphabet $\Sigma$. Is there a common name for the group $G$ presented as: $$G=\langle ...
2
votes
0answers
140 views

Algebras admitting quantifier elimination

I apologize if this question is meaningless or trivial: What are examples of Algebras admitting quantifier elimination? Especially are there Groups admitting quantifier elimination? I need to say ...
24
votes
1answer
2k views

A preprint of Sela concerning the work of Kharlampovich-Miyasnikov

Yesterday, Z. Sela published a preprint in arXiv which claims that the solution of Olga Kharlampovich and Alexi Miyasnikov for the Tarski problem on decidablity of the first order theories of free ...
4
votes
1answer
179 views

Why the axiomatic rank of the variety of groups is equal to three?

I am thankful of Anton Klyachko who introduced axiomatic rank to me: the axiomatic rank of a variety is the minimum number of variables which we need to define that variety by identities. It seems ...
0
votes
3answers
132 views

Negated varieties and their relatively free algebras

During the past days, I asked some questions in order to gain a clear understanding of the notion of "free algebras". I suppose that the question below is the most clear image of the concept I have ...
4
votes
3answers
248 views

The existence of an algebra whose set of identities and first order theory are equivalent

Is there an algebra $A$ (for example a group) such that $Th(A)$ is logically equivalent to $id(A)$? In other words, is there an algebra $A$ such that $$ Mod(Th(A))=Var(A)? $$ Clearly finite algebras ...
0
votes
1answer
148 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 ...
14
votes
2answers
1k views

Hilbert's 10th problem and nilpotent groups

I am asking this question on behalf of a colleague of mine who does not have an MO account. Nevertheless I am also interested in the answer. The question concerns relationships between Hilbert's ...
8
votes
0answers
168 views

A Magnus theorem in the category of residually finite groups

There is a natural notion of a presentations in the category of residually finite groups. Namely, if $X$ is set and $R$ is a set of words in the free group $FG(X)$ on $X$, then define $G=RF\langle ...
7
votes
2answers
351 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 ...
11
votes
3answers
2k 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. ...
8
votes
0answers
260 views

Automorphism group of the Turing degrees

It is conjectured that the automorphism group of the Turing degrees, $Aut(\mathcal{D})$, is trivial. However, to the best of my knowledge, the current state-of-the-art is that $Aut(\mathcal{D})$ is ...
5
votes
1answer
121 views

Free ultrafilters on groups and irregularity

Hello, Let $G$ be an infinite finitely generated discrete group. I call an infinite set $S$ irregular iff for every $g\in G$, $g\neq 1$, we have that $S\cap gS$ is finite. For example ...
2
votes
1answer
165 views

A categorical framework for Freiman s-morphisms

Let $\mathfrak A_i$ be groups ($i = 1, 2$), written multiplicatively, and $s$ a non-negative integer (here, as usual, I am abusing notation and denoting the operations of $\mathfrak A_1$ and ...
5
votes
0answers
133 views

Original source for the undecidabity of the first order theory of finite dimensional representations of the free algebra on 2 generators

Many books and papers on the representation theory of finite dimensional algebras state that the first order theory for finite dimensional modules for the free algebra on two generators is undecidable ...
7
votes
1answer
264 views

Membership problem for cyclic subgroups

Question: is there any example of a finitely presented (or at least finitely generated) group $G$ with an infinite cyclic subgroup $C \leqslant G$ such that the word problem in $G$ is solvable but ...
11
votes
1answer
419 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 ...
3
votes
1answer
327 views

Universe-sized groups with only set-sized normal subgroups, their cardinality in a certain range

Let $\kappa$ be an inaccessible cardinal, and let $G$ be a group with $|G| \geq \kappa$. For any cardinal $\lambda \le \kappa$ (regular, say, but not necessary), say $G$ is $\lambda$-simple if for all ...
0
votes
2answers
413 views

For the symmetric group on an infinite set, is there a generating set of strictly smaller cardinality? [closed]

Let $S_{\kappa}$ denote the symmetric group on some set of cardinality $\kappa$. Does there exist a generating set $X \subset S_{\kappa}$ such that $|X| < |S_{\kappa}|$ ($\stackrel{?}{=} ...
18
votes
3answers
742 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
votes
2answers
979 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 ...
10
votes
1answer
740 views

Cherlin's “Main Conjecture”

Cherlin's "Main Conjecture" from his 1979 paper "Groups of Small Morley Rank" is the following: Every simple $\omega$-stable group is an algebraic group over an algebraically closed field. Zilber was ...
15
votes
1answer
565 views

(Dis)similarity between groups and Lie algebras

There are many questions which sound similar or the same for groups and Lie algebras. Some (very few, actually) of those questions have identical solutions and answers. Some have identical answers but ...
15
votes
1answer
752 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 ...
17
votes
0answers
418 views

Density of first-order definable sets in a directed union of finite groups

This is a generalization of the following question by John Wiltshire-Gordon. Consider an inductive family of finite groups: $$ G_0 \hookrightarrow G_1 \hookrightarrow \ldots \hookrightarrow G_i ...
4
votes
1answer
275 views

What is known about the ultra-inverse limit?

Given a nonprincipal ultrafilter $\mu$ on $\mathbb{N}$ and a sequence of groups $G_i$, one can define its ultraproduct as: $$ ^*\prod_{i\in \mathbb{N}}G_i:=\{(x_i)_{i \in \mathbb{N}}| x_i\in ...
10
votes
3answers
736 views

Construction of a proper uncountable subgroup of $\mathbb{R}$ without Choice.

It is straightforward to construct proper uncountable subgroups of $\mathbb{R}$. One can construst a basis for $\mathbb{R}$ over $\mathbb{Q}$, and then there are many possibilities (just consider the ...
13
votes
1answer
432 views

Which finitely presented groups can be distinguished by decidable properties?

This question continues the line of inquiry of these three questions. Question. Which finitely presented groups can be distinguished by decidable properties? To be precise, let us say that φ is ...
11
votes
4answers
756 views

Does every decidable question about finitely presented groups amount to a question about abelian groups?

This question is about an issue left unresolved by Chad Groft's excellent question and John Stillwell's excellent answer of it. Since I find the possibility of an affirmative answer so tantalizing, I ...
18
votes
5answers
960 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 ...
6
votes
1answer
661 views

An ubiquitous pattern of questions

There is an ubiquitous pattern of questions concerning assumedly any kind of mathematical object or structure: groups, graphs, numbers, categories, and so on. It goes like this (informally): Can a ...
38
votes
5answers
4k views

Which graphs are Cayley graphs?

Every group presentation determines the corresponding Cayley graph, which has a node for each group element, and arrows labeled with the generators to get from one group element to another. My main ...
8
votes
1answer
198 views

How bad can the recursive properties of finitely presented groups be?

Any finitely presented group naturally gives rise to an edge-labeled graph (the Cayley graph) and I am considering paths through this graph. Paths correspond to infinite sequences of generators, so ...
56
votes
3answers
5k 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 ...
7
votes
2answers
538 views

elementary equivalence of infinitary symmetric groups

Two questions: Suppose a and b are two uncountable cardinals. Consider the symmetric groups on sets of sizes a and b respectively (the symmetric group on a set is the group of all bijections from ...