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

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In summary, K is a subgroup of G because it has all the necessary characteristics to be a subgroup of G, including an identity element, an inverse, and being closed.
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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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  • #2
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?
 
  • #3
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?
 
  • #4
Yep. (Nod)

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

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

What is subgroup nesting?

Subgroup nesting is a statistical method used to analyze data sets where groups are nested within larger groups. This method allows for the examination of both the within-group and between-group effects on a particular variable.

How is subgroup nesting different from other statistical methods?

Subgroup nesting differs from other statistical methods, such as ANOVA or regression, in that it takes into account the hierarchical structure of the data. This means that subgroup nesting allows for the examination of both the overall group effects as well as the effects of subgroups within the larger group.

What types of data sets are suitable for subgroup nesting analysis?

Subgroup nesting is suitable for data sets where groups are nested within larger groups, such as students within classrooms, employees within departments, or patients within hospitals. This method is particularly useful when the groups are not independent and when the group sizes vary.

What are the benefits of using subgroup nesting?

Subgroup nesting allows for a more nuanced analysis of data by accounting for the hierarchical structure of the data. It also allows for the examination of both group-level and individual-level effects, providing a more comprehensive understanding of the data.

What are some common challenges when using subgroup nesting?

One common challenge when using subgroup nesting is determining the appropriate level of analysis, as this can have a significant impact on the results. Additionally, subgroup nesting requires a larger sample size compared to other statistical methods, which can be a limitation in some cases.

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