The exclusion of empty substructures

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In summary, if the empty subset of a group G were considered a subgroup of G, then some theorems would become invalid.
  • #1
1MileCrash
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So, subspaces of vector spaces, and subgroups of groups, are not allowed to be empty.

This is because "there exists an identity element". We could include the empty set in these substructures but have the definition otherwise unchanged.

I'm curious as to what the consequences of such would be. If the empty subset of a group G were considered a subgroup of G, what would be some consequences in our important theorems?
 
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  • #2
All theorems which require a reference to the identity would have to include "except for the empty set", an unneeded complication.
 
  • #3
A subgroup is supposed to be a subset that's also a group. Since the empty set is not a group, it would be pretty odd to insist on calling it a subgroup.
 
  • #4
Fredrik said:
A subgroup is supposed to be a subset that's also a group. Since the empty set is not a group, it would be pretty odd to insist on calling it a subgroup.

Clearly, if the empty set were considered a subgroup, it would also be considered a group..
 
  • #5
1MileCrash said:
Clearly, if the empty set were considered a subgroup, it would also be considered a group..
OK. But that means that we would have to change the definition of "group" from

A pair ##(G,\star)## is said to be a group if ##\star## is a binary operation on ##G## that satisfies the group axioms.​

to

A pair ##(G,\star)## is said to be a group if ##G=\star=\varnothing## or ##\star## is a binary operation on ##G## that satisfies the group axioms.​

This doesn't look like an improvement.

Some theorems would remain intact. For example, consider the theorem "For all ##x,y,z\in G##, if ##x\star z=y\star z##, then ##x=y##." This statement is true when ##G=\varnothing##, because ##G## doesn't contain three elements ##x,y,z## such that the implication is false.
 
  • #6
Fredrik said:
OK. But that means that we would have to change the definition of "group" from

A pair ##(G,\star)## is said to be a group if ##\star## is a binary operation on ##G## that satisfies the group axioms.​

to

A pair ##(G,\star)## is said to be a group if ##G=\star=\varnothing## or ##\star## is a binary operation on ##G## that satisfies the group axioms.​

This doesn't look like an improvement.

Some theorems would remain intact. For example, consider the theorem "For all ##x,y,z\in G##, if ##x\star z=y\star z##, then ##x=y##." This statement is true when ##G=\varnothing##, because ##G## doesn't contain three elements ##x,y,z## such that the implication is false.

It's not really suggesting that it is an "improvement", I'm merely asking the question "what happens if we relax our axioms." We don't have to call this new object a group any more, it doesn't matter.

Immediately, Lagrange's Theorem will no longer work, for example, and G/{} would be a quotient group since {} is normal, and it seemingly would be the set {} again (the definition would lead {} to have no left cosets) but under the operation associated with quotient groups rather than that of G.So a lot of things get weird or break right off the bat, but I'm wondering if anything more interesting would arise.
 

FAQ: The exclusion of empty substructures

1. What is the exclusion of empty substructures?

The exclusion of empty substructures refers to the process of removing or not considering empty or null elements within a larger structure or system. This can be applied in various fields such as mathematics, computer science, and physics.

2. Why is it important to exclude empty substructures?

Excluding empty substructures is important because it allows for a more accurate and precise analysis of a system. By removing irrelevant or insignificant elements, we can focus on the essential components and better understand the behavior and properties of the structure.

3. How is the exclusion of empty substructures applied in mathematics?

In mathematics, the exclusion of empty substructures is used in various concepts such as set theory and graph theory. For example, when defining a set, we exclude the empty set as it does not contain any elements and does not contribute to the properties of the set.

4. Can the exclusion of empty substructures have practical applications?

Yes, the exclusion of empty substructures can have practical applications in fields such as data analysis and computer programming. By removing empty elements, we can reduce the amount of data to be processed, leading to more efficient algorithms and faster computations.

5. Are there any drawbacks to excluding empty substructures?

One potential drawback is that it may lead to oversimplification of a system. By excluding some elements, we may miss out on important information or relationships that could be crucial in understanding the overall structure. It is important to carefully consider which substructures to exclude and their potential impact on the analysis.

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