Skip to main content
MathOverflow

Return to Question

Post Undeleted by Stefan Kohl♦, Misha
Fixed a number of typos.
Source Link
Stefan Kohl ♦
Stefan Kohl ♦
  • 19.6k
  • 21
  • 75
  • 137

Suppose we have a triple of groups $G,H,K$ verifyingsatisfying the follwingfollowing conditions:

  1. $G$ and $H$ are finite groups and $K$ is an infinite group.
  2. there existsexist two monomorphisms $G\rightarrow K\leftarrow H$ inducing$G \rightarrow K \leftarrow H$ which induce an isomorphism in homology (with integral coefficients).

My question is the follwoingfollowing: ItIs there a knowknown triple $(K, G, H)$ verifiyingsatisfying the conditions 1(1.) and 2(2.) such that $G$ is not isomorphic to $H$  ? I'mI am more interested whenin the case that $G$ is a perfect group, but any example (if there exists such) is welcome.

Edit: In the case where $K$ is finite, CullerCuller's theorem says that $K, G, H$ are all isomorphic.

Suppose we have a triple of groups $G,H,K$ verifying the follwing conditions

  1. $G$ and $H$ are finite groups and $K$ an infinite group.
  2. there exists two monomorphisms $G\rightarrow K\leftarrow H$ inducing an isomorphism in homology (with integral coefficients)

My question is the follwoing: It there a know triple $(K, G, H)$ verifiying the conditions 1 and 2 such that $G$ is not isomorphic to $H$  ? I'm more interested when $G$ is a perfect group, but any example (if there exists) is welcome.

Edit: In the case where $K$ is finite, Culler theorem says that $K, G, H$ are all isomorphic.

Suppose we have a triple of groups $G,H,K$ satisfying the following conditions:

  1. $G$ and $H$ are finite groups and $K$ is an infinite group.
  2. there exist two monomorphisms $G \rightarrow K \leftarrow H$ which induce an isomorphism in homology (with integral coefficients).

My question is the following: Is there a known triple $(K, G, H)$ satisfying the conditions (1.) and (2.) such that $G$ is not isomorphic to $H$? I am more interested in the case that $G$ is a perfect group, but any example (if there exists such) is welcome.

Edit: In the case where $K$ is finite, Culler's theorem says that $K, G, H$ are all isomorphic.

Post Deleted by Ilias A.
added 96 characters in body
Source Link
Ilias A.
Ilias A.
  • 2k
  • 10
  • 18

Suppose we have a triple of groups $G,H,K$ verifying the follwing conditions

  1. $G$ and $H$ are finite groups and $K$ an infinite group.
  2. there exists two monomorphisms $G\rightarrow K\leftarrow H$ inducing an isomorphism in homology (with integral coefficients)

My question is the follwoing: It there a know triple $(K, G, H)$ verifiying the conditions 1 and 2 such that $G$ is not isomorphic to $H$ ? I'm more interested when $G$ is a perfect group, but any example (if there exists) is welcome.

Edit: In the case where $K$ is finite, Culler theorem says that $K, G, H$ are all isomorphic.

Suppose we have a triple of groups $G,H,K$ verifying the follwing conditions

  1. $G$ and $H$ are finite groups and $K$ an infinite group.
  2. there exists two monomorphisms $G\rightarrow K\leftarrow H$ inducing an isomorphism in homology (with integral coefficients)

My question is the follwoing: It there a know triple $(K, G, H)$ verifiying the conditions 1 and 2 such that $G$ is not isomorphic to $H$ ? I'm more interested when $G$ is a perfect group, but any example (if there exists) is welcome.

Suppose we have a triple of groups $G,H,K$ verifying the follwing conditions

  1. $G$ and $H$ are finite groups and $K$ an infinite group.
  2. there exists two monomorphisms $G\rightarrow K\leftarrow H$ inducing an isomorphism in homology (with integral coefficients)

My question is the follwoing: It there a know triple $(K, G, H)$ verifiying the conditions 1 and 2 such that $G$ is not isomorphic to $H$ ? I'm more interested when $G$ is a perfect group, but any example (if there exists) is welcome.

Edit: In the case where $K$ is finite, Culler theorem says that $K, G, H$ are all isomorphic.

Source Link
Ilias A.
Ilias A.
  • 2k
  • 10
  • 18

Finite groups inside an infinite group with the same homology

Suppose we have a triple of groups $G,H,K$ verifying the follwing conditions

  1. $G$ and $H$ are finite groups and $K$ an infinite group.
  2. there exists two monomorphisms $G\rightarrow K\leftarrow H$ inducing an isomorphism in homology (with integral coefficients)

My question is the follwoing: It there a know triple $(K, G, H)$ verifiying the conditions 1 and 2 such that $G$ is not isomorphic to $H$ ? I'm more interested when $G$ is a perfect group, but any example (if there exists) is welcome.