Does Preimage of Subgroup Under Homomorphism Form a Subgroup?

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Discussion Overview

The discussion revolves around whether the preimage of a subgroup under a homomorphism necessarily forms a subgroup. It explores the implications of subgroup criteria in the context of group theory.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant proposes that if θ: A → B is a homomorphism and S is a subgroup of B that is not completely contained in the range, then the preimage of S should form a subgroup.
  • Another participant mentions using the subgroup criterion test and concludes that the preimage should indeed be a subgroup, expressing a desire to confirm that no trivial details were overlooked.
  • A later reply agrees with the conclusion that the preimage is a subgroup, reinforcing the initial claim.

Areas of Agreement / Disagreement

While some participants express confidence that the preimage forms a subgroup, the discussion does not reach a consensus on whether this holds under all conditions, particularly regarding the containment within the range.

Contextual Notes

Participants do not clarify specific assumptions or conditions under which the preimage might fail to be a subgroup, leaving some uncertainty regarding the completeness of the argument.

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Suppose θ: A → B is a homomorphism. And assume S ≤ B. Is it necesarily true that if S is a subgroup, that is not completely contained in the range, its preimage forms a subgroup?
 
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Thoughts?
 
I used subgroup criterion test, and it should be a subgroup. But i just wanted to make sure I didn't miss anything trivial.
 
xiavatar said:
I used subgroup criterion test, and it should be a subgroup. But i just wanted to make sure I didn't miss anything trivial.

You didn't. It's a subgroup indeed!
 

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