Proving K is a Subgroup of G: Subgroup Nesting in H and L

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Discussion Overview

The discussion centers on proving that a set K is a subgroup of a group G, given that H is a subgroup of G and L is a subgroup of H. The scope includes theoretical aspects of group theory and subgroup properties.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • Some participants express that the question seems redundant, suggesting that anything in a subgroup should automatically be a subgroup of any larger group that contains it.
  • One participant questions if there is a typo in the problem, proposing that K and L might be the same set.
  • Another participant emphasizes the importance of applying the definition of a subgroup to verify that K meets all necessary conditions to be a subgroup of G.
  • A participant outlines a reasoning process: K is nonempty, contains an identity element, includes inverses for any element, and is closed under the group operation, leading to the conclusion that K is a subgroup of G.
  • There is a correction regarding the language used to describe the inverses, suggesting it should refer to 'any' element rather than 'an' element.

Areas of Agreement / Disagreement

Participants generally agree on the necessity of applying subgroup definitions, but there is no consensus on the redundancy of the question or the interpretation of K and L.

Contextual Notes

The discussion does not resolve whether K and L are indeed the same set, and the implications of this assumption remain unclear.

Jen917
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Let H be a subgroup of G and let L be a subgroup of H. Prove that K is a subgroup of G.

This question seems very redundant to me, isn't anything in a subgroup automatically a subgroup of anything the larger group is a subgroup of. Can some one explain this proof to me?
 
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Jen917 said:
Let H be a subgroup of G and let L be a subgroup of H. Prove that K is a subgroup of G.

This question seems very redundant to me, isn't anything in a subgroup automatically a subgroup of anything the larger group is a subgroup of. Can some one explain this proof to me?

Hi again Jen917! (Wave)

Can I assume there is a typo, and that K and L are actually the same?

Then admittedly, the proof is pretty straight forward.
It's just that in math we can't assume that it's redundant.

To conclude that one set is a subgroup of another, we have to apply the definition of a subgroup, and verify if all conditions are fulfilled.
This would be an exercise in carefully reading and applying a definition.
Which proof do you have?
 
I like Serena said:
Hi again Jen917! (Wave)

Can I assume there is a typo, and that K and L are actually the same?

>>Yes! Sorry about the typo.

Then admittedly, the proof is pretty straight forward.
It's just that in math we can't assume that it's redundant.

>>Gotcha, I think I was overthinking it. The easy answer seemed just too easy!

To conclude that one set is a subgroup of another, we have to apply the definition of a subgroup, and verify if all conditions are fulfilled.
This would be an exercise in carefully reading and applying a definition.
Which proof do you have?

This was my idea:
K is a nonempty set and has an identity element, we know this because it is a subgroup of H.
K also contains the inverse of an element following the same logic.
Finally we know K is closed because it is a subgroup of H.
H is a subgroup of G, therefore K has all these characteristics within G and is a subgroup of G.

Is this the along the right idea?
 
Yep. (Nod)

Nitpick: the sentence about the inverse should be about 'any' element rather than 'an' element.
 

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