All Questions
Tagged with model-theory group-theory
77
questions
9
votes
2
answers
457
views
If $a$ and $b$ are elements of a group $G$ that satisfy the same first order formulas, is there always an automorphism of $G$ that maps $a$ to $b$?
Let $a$ and $b$ be elements of group $G$ and assume that for any natural number $n$ and for any first-order formula $\varphi (x_1,x_2,...,x_n)$ with $n$ free variables in the language of groups, $G \...
- 446
4
votes
2
answers
155
views
Describe the class of groups that satisfy $(x^2y^2)^2 \approx 1$
I have trouble with the following assignment:
Let $A$ denote the class of all Abelian groups satisfying the identity $x^2 \approx 1$. Show that the class $$\{G \mid \exists N: N \trianglelefteq G \...
- 140
3
votes
0
answers
69
views
G is a connected omega-stable group of finite Morley rank then its derived group is definable and connected
I am reading Lascar's article "Omega-stable groups" in the book of Bouscaren "Model Theory and Algebraic Geometry". In section 5, he applies the indecomposability theorem to show ...
- 384
0
votes
2
answers
190
views
How a theorem in a mathematical theory can be applied to a model of the mathematical theory?
Jean Dieudonne, one of the most prominent
mathematicians of the 20th century, stated[1. p.215] that any mathematical theory is an extension of ZF set theory:
"The theory of sets, so conceived, ...
- 617
0
votes
1
answer
77
views
Is the class of all symmetric groups an elementary class? [closed]
I define a symmetric group to be a group isomorphic to the group of all bijections on some set $S$. $S$ does not have to be finite, it could be infinite. My question is, is the class $K$ of all ...
- 21.4k
1
vote
1
answer
269
views
prove that none of the following theories is finitely axiomatizable:
prove that none of the following theories is finitely axiomatizable:
$(a)$ infinite models of sets with only equality;
$(b)$ fields of characteristic zero;
$(c)$ divisible Abelian groups;
$(d)$ ...
- 131
2
votes
1
answer
175
views
G is closed under elementary equivalence (use Keisler–Shelah Isomorphism theorem)
Been trying to solve the following logic question:
Show that the class:
$\mathscr G$ = {$G$ is a group| $G\subseteq$ $SL_n$($\mathbb F$), where n $\in$ $\mathbb N$, $\mathbb F$ is any field}
is ...
- 131
4
votes
1
answer
114
views
Satisfiability of Divisible Group Axioms
I'm struggling with this problem:
Let $\mathcal{G}=(G,+,-,0)$ be an abelian group. Work in the language $L=\{+,-,c_0\}$ and expand it with a constant symbol $c_g$, for each $g\in G\setminus \{0\}$. ...
- 391
1
vote
0
answers
95
views
Inexpressiblity of a group having an element of infinite order
Let $\mathcal{L} = \{ e, \cdot \}$ be the first-order language of groups.
I want to show that the property of a group having an element of infinite order is inexpressible in $\mathcal{L}$. I haven't ...
1
vote
1
answer
100
views
Is the class of modular/distributive groups an axiomatizable class? [closed]
I define a modular group to be a group whose lattice of subgroups is a modular lattice, and similarly for distributive groups. My question is, are either of modular and/or distributive groups a first-...
- 21.4k
5
votes
2
answers
75
views
Is $SL(2, \mathbb{R})$ dense in saturated elementarily extensions of the reals?
Suppose $\mathcal{A} = \langle A, <, +, \cdot \rangle$ is a $\aleph_1$-saturated elementarily extension of the real field.
Is $SL(2, \mathbb{R})$ dense in $SL(2,A)$?
In compact groups one can find ...
- 601
2
votes
1
answer
116
views
The compactness theorem for simple groups
I would like to use the compactness theorem to prove the following:
Claim. If $H$ is a subgroup of countable index in a simple group $G$, then there are finitely many conjugates of $H$ with trivial ...
- 601
0
votes
1
answer
72
views
Disjoint union of copies of Henson graph is not homogenous
From Wiki, the Henson graph $G_i$ is an undirected infinite graph, the unique countable homogeneous graph that does not contain an i-vertex clique but that does contain all $K_i$-free finite graphs as ...
- 1
3
votes
1
answer
148
views
When is the group-theoretic notion of algebraic closure idempotent?
In model theory, the notion of algebraic closure can be defined in terms of formulas having only finitely many realizers. Alternatively, it can be defined in terms of automorphisms: $a \in M$ is in ...
- 7,290
8
votes
1
answer
169
views
Is there a model of $Th(\frac{\mathbb{R}}{\mathbb{Z}})$ which is a periodic group?
Let $L := (+,-,0)$ be the language of abelian groups and let $T = Th(\frac{\mathbb{R}}{\mathbb{Z}})$. Is there a model of $T$ which is periodic, i.e. every member of the domain has finite order. In ...
- 105