# Tagged Questions

**3**

votes

**2**answers

473 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

**0**answers

143 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

**1**answer

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

**3**answers

133 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

**3**answers

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

**1**answer

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 ...

**49**

votes

**4**answers

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

**1**answer

185 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

**0**answers

153 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

**2**answers

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

**2**answers

687 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

**1**answer

743 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

**1**answer

633 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

**6**answers

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 ...