Skip to main content
MathOverflow

Return to Question

deleted 313 characters in body; edited title
Source Link
YCor
YCor
  • 63.9k
  • 5
  • 187
  • 286

Countably infinite groups Number of "character $0$"countable torsion-free groups

Let us say a group $(G,\cdot)$ has character $0$ if for all $g\in G\setminus\{e\}$ and for all positive integers $n$ we have $g^n \neq e$ (where $e$ is the neutral element of the group.

Is it true that there a collectionexist ${\cal G}$ of countably infinite$2^{\aleph_0}$ pairwise non-isomorphic torsion-free countable groups, such that

  1. each $G\in {\cal G}$ has character $0$,
  2. if $G_1\neq G_2\in{\cal G}$ then $G_1\not\cong G_2$, and
  3. $|{\cal G}| = 2^{\aleph_0}$?

?

Countably infinite groups of "character $0$"

Let us say a group $(G,\cdot)$ has character $0$ if for all $g\in G\setminus\{e\}$ and for all positive integers $n$ we have $g^n \neq e$ (where $e$ is the neutral element of the group.

Is there a collection ${\cal G}$ of countably infinite groups, such that

  1. each $G\in {\cal G}$ has character $0$,
  2. if $G_1\neq G_2\in{\cal G}$ then $G_1\not\cong G_2$, and
  3. $|{\cal G}| = 2^{\aleph_0}$?

?

Number of countable torsion-free groups

Is it true that there exist $2^{\aleph_0}$ pairwise non-isomorphic torsion-free countable groups?

Source Link
Dominic van der Zypen
Dominic van der Zypen
  • 48.9k
  • 8
  • 47
  • 162

Countably infinite groups of "character $0$"

Let us say a group $(G,\cdot)$ has character $0$ if for all $g\in G\setminus\{e\}$ and for all positive integers $n$ we have $g^n \neq e$ (where $e$ is the neutral element of the group.

Is there a collection ${\cal G}$ of countably infinite groups, such that

  1. each $G\in {\cal G}$ has character $0$,
  2. if $G_1\neq G_2\in{\cal G}$ then $G_1\not\cong G_2$, and
  3. $|{\cal G}| = 2^{\aleph_0}$?

?