All Questions
Tagged with gr.group-theory lo.logic
105 questions
8
votes
1
answer
312
views
Elementary extensions of direct product
I apologize if this question is elementary: Let $A$, $B$, and $A^{\prime}$ be groups such that $A^{\prime}$ is an elementary extension of $A$. Is it true that $A^{\prime}\times B$ is an elementary ...
- 2,233
24
votes
8
answers
3k
views
Applications of logic to group theory?
There seems to be an ever-growing literature on the first-order theory of groups. While I find this interaction between group theory and logic quite appealing, I was wondering the following:
Are ...
- 359
35
votes
2
answers
3k
views
Is Lagrange's Theorem equivalent to AC?
Lagrange's Theorem is most often stated for finite groups, but it has a natural formation for infinite groups too: if $G$ is a group and $H$ a subgroup of $G$, then $|G| = |G:H| \times |H|$.
If we ...
- 643
6
votes
1
answer
226
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 ...
- 1,033
12
votes
2
answers
621
views
Eliminating constant in Rado graph
Let $R$ denote the Rado graph, and let $c$ be a fixed vertex.
Question 1. Is the structure obtained by extending $R$ by the constant $c$ interpretable in $R$ without parameters?
By interpretable I ...
- 618
10
votes
2
answers
611
views
Normal subgroups of automorphism group of relational structure
Let $S$ be the set of all finite permutations of $\mathbb{N}$, i.e. they fix all but a finite set, and $A\subset S$ the set of all even permutations.
Theorem. The normal subgroups of $S_\infty$ are ...
- 2,882
12
votes
0
answers
468
views
A question concerning model theory of groups
Several days ago, Professor Martin Bridson gave a very nice talk in my department. Some questions concerning his talk came into my brain Since I am neither a model theorist nor a algebraist, I am not ...
- 4,201
2
votes
0
answers
90
views
Fully residually free groups and completion
Let $G$ be a fully residually free group with a finitely generated profinite completion. Is $G$ necessarily finitely generated?
- 11.3k
3
votes
1
answer
374
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 $...
- 11.3k
3
votes
2
answers
290
views
Do limit groups satisfy Howson's theorem?
Let $G$ be a limit group, and let $A,B \leq G$ be finitely generated
subgroups generating $G$ (i.e. $\langle A \cup B \rangle = G$). Must
$A \cap B$ be finitely generated?
Recall that a limit ...
- 11.3k
9
votes
1
answer
695
views
Extending an infinite simple group
Maybe the question does not fit here.
Yesterday in my logic course, I presented a nice example about an application of model theory to group theory. The example is due to Hodges and as following: For ...
- 4,201
5
votes
0
answers
307
views
Proving that a subgroup is normal
This question is partly inspired by the recent question on measurability and the axiom of choice.
Suppose I come up with a way to define a subgroup of a group, in a way that involves "no arbitrary ...
- 82.7k
6
votes
1
answer
1k
views
Countable group with uncountable number of subgroups $< 2^{\aleph_0}$ [duplicate]
Is it consistent that there is a countable group $G$ such that the cardinality of the set of subgroups of $G$ is uncountable, but strictly less than $2^{\aleph_0}$?
- 48.9k
13
votes
0
answers
387
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
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
6
votes
1
answer
1k
views
Elementary equivalence of the direct product and direct sum of groups
It is well-known that the direct product of any family of abelian groups
is an elementary extension of the direct sum of the family
(see e.g. Lemma A.1.6 in the book `Model Theory' by W. Hodges,
...
- 893
7
votes
3
answers
678
views
Decision problem on triviality of intersection of two subgroups
What is known about the following decision problem?
Given two finite sets in a finitely generated group G,
decide whether the subgroups generated by them have trivial intersection.
Is this problem ...
- 893
11
votes
2
answers
778
views
History of Tarski's problems on free groups
As is known, Tarski posed his questions about first-order theories of non-abelian free groups around 1945. However, the questions were not published in his papers or books.
What is the original ...
- 893
5
votes
1
answer
884
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 ...
- 829
84
votes
3
answers
6k
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 ...
- 29.8k
6
votes
2
answers
490
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:
$$G=\langle ...
- 2,233
2
votes
1
answer
330
views
Algebras admitting quantifier elimination
I apologize if this question is meaningless or trivial:
What are examples of Algebras admitting quantifier elimination? Especially are there Groups admitting quantifier elimination?
I need to say ...
- 2,233
25
votes
1
answer
3k
views
A preprint of Sela concerning the work of Kharlampovich-Miyasnikov
Yesterday, Z. Sela published a preprint in arXiv which claims that the solution of Olga Kharlampovich and Alexi Miyasnikov for the Tarski problem on decidablity of the first order theories of free ...
- 2,233
4
votes
1
answer
206
views
Why the axiomatic rank of the variety of groups is equal to three?
I am thankful of Anton Klyachko who introduced axiomatic rank to me: the axiomatic rank of a variety is the minimum number of variables which we need to define that variety by identities.
It seems ...
- 2,233
0
votes
3
answers
185
views
Negated varieties and their relatively free algebras
During the past days, I asked some questions in order to gain a clear understanding of the notion of "free algebras". I suppose that the question below is the most clear image of the concept I have ...
- 2,233
5
votes
3
answers
309
views
The existence of an algebra whose set of identities and first order theory are equivalent
Is there an algebra $A$ (for example a group) such that $Th(A)$ is logically equivalent to $id(A)$? In other words, is there an algebra $A$ such that
$$
Mod(Th(A))=Var(A)?
$$
Clearly finite algebras ...
- 2,233
2
votes
4
answers
555
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 ...
- 2,233
13
votes
2
answers
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 ...
- 65.4k
8
votes
0
answers
298
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 X\...
- 38.6k
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 ...
- 39.8k
39
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. ...
- 1,687
6
votes
3
answers
872
views
An element $g$ in a group such that neither $g=1$ nor $g\ne 1$ can be proved.
Edited (this question contains two versions of a similar question)
Is there some finitely presented group $G$ generated by $g_1,...,g_n$ such that
there is an element $g\in G$ expressed as a finite ...
- 14.3k
18
votes
1
answer
921
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 ...
- 20.5k
6
votes
1
answer
186
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 $\{z^3|z\in\...
- 1,550
2
votes
1
answer
220
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 $\...
- 10.5k
5
votes
0
answers
154
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 ...
- 38.6k
8
votes
1
answer
638
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 ...
- 3,181
14
votes
1
answer
953
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 ...
- 31.2k
3
votes
1
answer
383
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
2
answers
963
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{?}{=} 2^{\kappa}$...
- 19
21
votes
4
answers
2k
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 ...
- 25k
30
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 ...
- 18.7k
11
votes
1
answer
1k
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 ...
- 359
18
votes
1
answer
727
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 ...
- 2,804
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).$$
This is the $n$th Laver table $(A_n,...
- 3,547
21
votes
0
answers
578
views
Density of first-order definable sets in a directed union of finite groups
This is a generalization of the following question by John Wiltshire-Gordon.
Consider an inductive family of finite groups:
$$
G_0 \hookrightarrow G_1 \hookrightarrow \ldots \hookrightarrow G_i \...
- 2,200
5
votes
1
answer
345
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 G_i\}/\...
- 11.1k
15
votes
3
answers
2k
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 ...
- 2,441
17
votes
1
answer
875
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 ...
- 236k
19
votes
4
answers
1k
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 ...
- 236k