Normal subgroup; topological group

In summary, the quotient R/Z is a familiar topological group that can be identified as a quotient topology of the real line where x and y are equivalent if they differ by an element of Z. This is a good choice for a set of equivalence classes of cosets.
  • #1
tomboi03
77
0
The integers Z are a normal subgroup of (R,+). The quotient R/Z is a familiar topological group; what is it?

okay... i attempted this problem...
and I don't know if i did it right... but can you guys check it?

Thanks~

R/Z is a familiar topological group
and Z are a normal subgroup of (R,+)
then the quotient R/Z is also a topological group with the quotient topology.
The R/Z is Hausdorff iff Z is a closed subset of (R,+) which in this case is true.
Therefore, R/Z is a quotient topology.

sorry if there are a lot of things that didn't make sense in that proof but i tried... hahaha
 
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  • #2
That's not what it's after. It's asking you to identify it (name it, if you will) as a topological group. What topological groups do you know?

You're taking the real line and identifying x and y if x-y is what? I.e. x and y differ by what? What is a good choice for a set of equivalence classes of cosets?
 

1. What is a normal subgroup?

A normal subgroup is a subgroup of a group that is invariant under conjugation by elements of the larger group. This means that if an element of the larger group is conjugated with an element of the normal subgroup, the result will still be in the normal subgroup.

2. How is a normal subgroup related to a topological group?

A topological group is a group that is also a topological space, meaning that it has a defined set of open sets. A normal subgroup in a topological group is a subgroup that is also a closed set, meaning that it contains all of its limit points. This allows for the study of the topological properties of the group as well as its algebraic structure.

3. Are all subgroups of a topological group normal?

No, not all subgroups of a topological group are normal. A subgroup is only considered normal if it is invariant under conjugation by elements of the larger group. If this condition is not met, then the subgroup is not considered normal.

4. How can normal subgroups be used in the study of topological groups?

Normal subgroups play a crucial role in the study of topological groups as they help to define the quotient group, which is the group formed by the cosets of the normal subgroup. This allows for the analysis of the group's topological and algebraic properties separately, making it easier to understand the overall structure of the group.

5. Can a normal subgroup also be a topological group?

Yes, a normal subgroup can also be a topological group. This can happen if the normal subgroup has a compatible topology with the larger group, meaning that the group operations and the topology are compatible with each other. In this case, the normal subgroup can be considered a topological group in its own right.

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