Finding middle group in short exact sequence

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Discussion Overview

The discussion revolves around the determination of the middle group in a short exact sequence, specifically the sequence 0->Z->A->Z_4->0. Participants explore methods for identifying the middle group without knowledge of the maps involved, discussing theoretical implications and examples.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants suggest that by the first isomorphism theorem, A/Z is isomorphic to Z_4, implying that A can be identified if a group with Z as a subgroup exists such that the quotient gives Z_4.
  • Others mention that A is an extension of Z by Z_4, with examples including Z x Z_4, but note that there could be other possibilities as well.
  • It is proposed that there are infinitely many groups that can occupy the middle position in the short exact sequence, corresponding to elements of Ext^1(Z_4,Z), which is an Abelian group.
  • Some participants indicate that if the Ext group is zero, then the middle term must be Z x Z_4, but if not, there are many more possibilities.
  • A request is made for additional examples of groups that could fit in the middle of the sequence.

Areas of Agreement / Disagreement

Participants generally agree that there are multiple possibilities for the middle group, but there is no consensus on specific examples beyond those mentioned. The discussion remains unresolved regarding the exact nature of the middle group without further information.

Contextual Notes

Participants note that the determination of the middle group is dependent on the properties of the Ext^1 group and the specific structure of the groups involved, which may not be straightforward to compute.

wodhas
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Hi,

Is there any method by which problems involve short exact sequences can be solved.

For example, if I have :

0->Z->A->Z_4->0

how can I decide what is the middle group without knowing any of the maps

Thanks!
S.
 
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Well, by the first isomorphism theorem, we have, modulo isomorphisms, that A/Z = Z_4. So if you can find a group having Z as a subgroup and such that when quotiented by Z gives Z_4, then A will be isomorphic to that...
 
0->Z->A->Z_4->0

Then A is called an extension of Z by Z_4 ... one example is the product Z x Z_4, but there could be others as well. The collection of all such extensions is sometimes given some structure and made into an algebraic object itself ... look up Ext and Tor I don't remember the details.
 
wodhas said:
Hi,

Is there any method by which problems involve short exact sequences can be solved.

For example, if I have :

0->Z->A->Z_4->0

how can I decide what is the middle group without knowing any of the maps

Thanks!
S.

In general you can not tell. But you can narrow down the possibilities. As other replies pointed out the middle group in your example could be Z or it could be ZxZ/4.
 
In general there are infinitely many groups that can go in the middle of a short exact sequence. In fact they are in one to one correspondence with elements of Ext^1(Z_4,Z) (which is an Abelian group). Now if this Ext group is zero, then you know for sure, the middle term must be ZxZ/4, otherwise there are many more possibilities. To compute this Ext^1 group look at the long exact sequence of cohomology groups arising from your short exact sequence or try to construct some resolutions. I'm not sure how hard either one of those tasks are in practice.(If you aren't sure what that means I recommend looking it up Ext on wikipedia)
 
Tzar said:
In general there are infinitely many groups that can go in the middle of a short exact sequence. In fact they are in one to one correspondence with elements of Ext^1(Z_4,Z) (which is an Abelian group). Now if this Ext group is zero, then you know for sure, the middle term must be ZxZ/4, otherwise there are many more possibilities. To compute this Ext^1 group look at the long exact sequence of cohomology groups arising from your short exact sequence or try to construct some resolutions. I'm not sure how hard either one of those tasks are in practice.(If you aren't sure what that means I recommend looking it up Ext on wikipedia)

Can you give me five other possibilities?
 

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