3
votes
2answers
467 views

A double centralizing theorem for finite groups

I have a proof for the following assertion which employs Model Theory. It has certainly a pure group theoretic proof, but what is such a proof? Is the assertion trivial? Theorem Let $G$ be a finite ...
2
votes
0answers
142 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 ...
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
249 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
149 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 ...
48
votes
4answers
2k views

Is there a 0-1 law for the theory of groups?

Several months ago, Dominik asked the question Is there a 0-1 law for the theory of groups? on mathstackexchange, but although his question received attention there is still no answer. By asking the ...
2
votes
1answer
173 views

Every abelian torsion-free group is strictly totally orderable (via the compactness theorem)

Let $\mathbb G = (G, +)$ be a group. We say that $\mathbb G$ is strictly totally orderable (others would say bi-orderable) if there exists a total order $\preceq$ on $G$ such that $x+z \prec y + z$ ...
2
votes
0answers
152 views

Is this fragment of arithmetic on $p^{-\infty} {\mathbb Z}$ decidable?

Let $p$ be a prime number. Consider the abelian group $p^{-\infty} {\mathbb Z} = \bigcup p^{-n} {\mathbb Z}$ consisting of rational numbers whose denominator is a power of $p$, under addition. View ...
4
votes
2answers
319 views

On the theory of infinite extraspecial $p$-groups

$p$ is a prime number. A group $G$ is called an infinite extraspecial $p$-group if 1) it is infinite, 2) every $g\neq 1$ in $G$ has order $p$, 3) its centre $Z(G)$ coincide with $G'$ and is a ...
12
votes
2answers
686 views

Connes' embedding conjecture for uncountable groups

In this topic, I will use the word uncountable group referring to groups whose cardinality is $\leq|\mathbb R|$. Notation: $R$ is the hyperfinite $II_1$-factor, $\omega$ is a free ultrafilter on the ...
10
votes
1answer
741 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 ...
6
votes
1answer
632 views

subgroups of the direct product intersecting trivially with the direct sum

Suppose I take an infinite direct power $\prod G$ of some (not necessarily finite) group $G$. I want to know about the subgroups of $\prod G$ that are maximal subject to having trivial intersection ...
15
votes
6answers
1k views

Three questions on large simple groups and model theory

Yesterday, in the short course on model theory I am currently teaching, I gave the following nice application of downward Lowenheim-Skolem which I found in W. Hodges A Shorter Model Theory: Thm: Let ...