Is it Possible to Embed F/R into F without Complementation?

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Discussion Overview

The discussion centers around the possibility of embedding the quotient group F/R into the group F without the need for complementation of the normal subgroup R. Participants explore theoretical implications, conceptual understandings, and examples related to group theory.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question why such an embedding would exist, suggesting that conceptually it does not make sense.
  • Examples are provided, such as the embedding of Z/2Z into Z, to illustrate points about embeddings and normal subgroups.
  • One participant argues that embedding F/R into F implies that R must be complemented in F by another subgroup, drawing parallels to vector spaces.
  • There is a discussion about the necessity of the canonical exact sequence splitting for the embedding to occur, with references to semidirect products.
  • Participants express skepticism about the assumption that every normal subgroup of F has the property of being complemented.
  • Clarifications are sought regarding the meaning of "split" in the context of the exact sequence.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there are multiple competing views regarding the existence of such embeddings and the conceptual framework surrounding them.

Contextual Notes

Some limitations include the dependence on definitions of complementation and the specific properties of normal subgroups, which remain unresolved in the discussion.

tgt
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Why doesn't such an embedding exist? When R is a normal subgroup of F.
 
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Why would such an embedding exist? Conceptually it doesn't make sense.

By way of example, can you embed Z/2Z into Z?
 
morphism said:
Why would such an embedding exist? Conceptually it doesn't make sense.

By way of example, can you embed Z/2Z into Z?

Sure, I see the exampe but can you explain the conceptual part? How is it conceptual to you?
 
Well, if you think about it for a bit, what does it mean to be able to embed F/R into F? Intuitively, modding out by a subgroup means you collapse that subgroup to zero. So for F/R to be isomorphic to some subgroup of F, it would be necessary for R to be 'complemented' inside of F by another subgroup (think of complementation here as you would in the setting of vector spaces). I personally don't see any reason why one would believe every normal subgroup of F to have this property. (But I admit to being prejudiced: functional analysis has made me very suspicious of the process of complementation!)

To make this a bit more precise, we can use semidirect products: If F/R embeds into F, then F is essentially [itex]R \rtimes F/R[/itex]. It's easy to see that a necessary and sufficient condition for this to happen is that the canonical exact sequence

[tex]1 \longrightarrow R \longrightarrow F \longrightarrow F/R \longrightarrow 1[/tex]

splits.

By the way, is there any specific reason you're using F and R to denote groups?!
 
morphism said:
Well, if you think about it for a bit, what does it mean to be able to embed F/R into F? Intuitively, modding out by a subgroup means you collapse that subgroup to zero. So for F/R to be isomorphic to some subgroup of F, it would be necessary for R to be 'complemented' inside of F by another subgroup (think of complementation here as you would in the setting of vector spaces). I personally don't see any reason why one would believe every normal subgroup of F to have this property. (But I admit to being prejudiced: functional analysis has made me very suspicious of the process of complementation!)

To make this a bit more precise, we can use semidirect products: If F/R embeds into F, then F is essentially [itex]R \rtimes F/R[/itex]. It's easy to see that a necessary and sufficient condition for this to happen is that the canonical exact sequence

[tex]1 \longrightarrow R \longrightarrow F \longrightarrow F/R \longrightarrow 1[/tex]

splits.

By the way, is there any specific reason you're using F and R to denote groups?!

Some good points raised. Although what does split mean?

Those notations naturally fit into the notation of a presentation.
 
Last edited:

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