6
votes
1answer
109 views

Elementary equivalence of free groups

This must be known inside out by model theorists by I have no cluse whether the following is true or not: Denote by $F_n$ the free group on $n$ generators. Suppose that $n\neq m$. Are the groups ...
2
votes
0answers
39 views

Counter example of monotone union [duplicate]

I saw this exercise in "Elements of Abstract and Linear Algebra" by E. H. Connell: Suppose $G$ is a group. Suppose $T$ is an index set and for each $t \in T$, $H_t$ is a subgroup of $G$. Furthermore, ...
3
votes
1answer
68 views

isomorphism between divisible, totally ordered, abelian groups

Let $G$, $H$ be divisible, abelian, linearly ordered groups, whose cardinalities are equal and satisfy $\mu := |G|=|H|>\aleph_{0}$. These are supposed to be (order!) isomorphic. And just about ...
2
votes
2answers
68 views

extension of group operation from $\mathbb{Q}$ to $\mathbb{R}$

I'm having a hard time with this (seems easy, but could be misleading) problem: Let $A \subseteq \mathbb{Q}$ be a convex subset, and let $+$ group operation on $A$. Let $\overline{A} := \{x \in ...
4
votes
1answer
133 views

First order theory of abelian groups and first order theory of cyclic groups are coincide?

Let $T$ be a first-order theory of cyclic groups. Even if an abelian group $(G,+)$ satisfy $(G,+)\models T$ there is no reason that $(G,+)$ is a cyclic. (For example, by Löwenheim–Skolem theorem there ...
5
votes
3answers
172 views

Showing that the Class of Cyclic Groups Aren't Axiomatizable

The class of finite cyclic groups are not axiomatizable, for if we supposed they were by some set of sentences $\Sigma$, then there would exist a model for $\Sigma$ of at least order $n$ for all $n ...
3
votes
2answers
110 views

Does the language of group theory need a constant?

I would like to verify the understanding of my claim. In Model Theory: An Introduction by David Marker Example 1.2.5 defines the language of the theory of groups as follows to be $\mathcal{L}=\{., ...
6
votes
1answer
113 views

Can there be an abelian group $G$ where $\bigcap_{n\in\mathbb N} nG$ is not divisible?

I heard a talk yesterday on MacIntyre's theorems, which involved a decomposition of ($\omega$-stable) groups into a divisible part and a bounded exponent part. Apparently there is a result of ...
3
votes
1answer
135 views

Number of isomorphism classes of countable models of a theory

Whether there are countably or uncountably many isomorphism classes of countable models of a given theory depends on the theory: if the theory is strong enough, there will be only countably many ...
130
votes
1answer
3k views

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

For each first order sentence $\phi$ in the language of groups, define : $$p_N(\phi)=\frac{\text{number of nonisomorphic groups $G$ of order} \le N\text{ such that } \phi \text{ is valid in } ...
0
votes
1answer
149 views

first-order logic: Any finite set of first-order sentences true in all divisible abelian groups is true in some non-divisable one.

The question is the Subject. This comes from Barwise: Intro to First Order Logic, it is prop 2.1 and it is answered 2 pages later. I however don't understand the proof well. As Barwise says, the ...
4
votes
1answer
67 views

Class of finite groups a Fraïssé Class? [duplicate]

Is the class of finite groups a Fraïssé class? Calling this class $K$, does $K$ satisfy the following: Joint embedding property Amalgamation property Hereditary property: if $G \in K$ and $H \le G$, ...
6
votes
2answers
173 views

Model theory in group theory

I am interested in useful results for group theorists that can be shown using model theory. For example : Theorem: Let $\langle X \mid R \rangle$ be presentation of a group $G$ with $X$ finite and ...
4
votes
0answers
109 views

is a group that eliminates quantifiers in the group language Abelian?

Let $G$ be a group. Consider it as a structure in the language of groups $L=(e,\cdot, (-)^{-1})$. Suppose that $G$ eliminates quantifiers in this language. Is it true that $G$ must be Abelian then? ...
4
votes
1answer
257 views

how to show that a group is elementarily equivalent to the additive group of integers

Is there any fairly easy way of showing a group is elementarily equivalent to the additive group of the integers? I've found a simple characterization here: A ‘natural’ theory without a prime model, ...
8
votes
2answers
236 views

Fraïssé limits and groups

I was recently reading up on Fraïssé limits in Hodges' "A shorter model theory." I was trying to think of some examples and wanted to see if I could take the Fraïssé limit on the category of finite ...