Exploring the Connection Between Cosets and Normal Subgroups

In summary, The conversation discusses the relationship between cosets and normal subgroups in a group. A (left) coset partitions a group, just like a normal subgroup does. However, there is a deeper connection between the two. A normal subgroup is a subgroup whose left cosets are also right cosets, and the cosets of a normal subgroup form a group on their own, called the quotient group. The speaker is seeking clarification on the specific relationship between cosets and normal subgroups and the meaning of normal subgroups partitioning the group.
  • #1
walwaldoggy
1
0
Hey guys
I'm curious about how to interpret cosets and normal subgroups.
I do know the definitions of both, but I do not understand how they relate to each other.
A (left) coset is supposed to partition a group as well as normal subgroups, but I'm sure there is a more profound relationship between the two than just the similarity I have pointed out.

I'll look forward to your inputs!
 
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  • #2
I'm not sure what you're looking for. A normal subgroup is simply a subgroup who's left cosets are also right cosets. Also the cosets of a normal subgroup form a group themself: the quotient group.

There are certainly nice relationships between cosets and normal subgroups, but I'm not certain which ones you want. Maybe you can clarify?

Also, I'm not quite sure what you mean with "normal subgroups partition the group"...
 

1. What is a coset?

A coset is a subset of a group that is created by multiplying each element of the subset by a fixed element of the group. It is represented by the notation aH, where a is the fixed element and H is the subset.

2. How is a coset different from a subgroup?

A subgroup is a subset of a group that is also a group itself, while a coset is a subset that is not necessarily a group. Subgroups have the same identity element as the original group, while cosets may have a different identity element.

3. What is a normal subgroup?

A normal subgroup is a subgroup that is invariant under conjugation by all elements of the original group. This means that if g is an element of the original group and h is an element of the normal subgroup, then ghg^-1 is also an element of the normal subgroup.

4. How do we determine if a subgroup is normal?

A subgroup is normal if and only if it is the kernel of a homomorphism from the original group to another group. In other words, if the subgroup is the set of elements that map to the identity element in the other group.

5. What is the significance of normal subgroups?

Normal subgroups play an important role in the structure of groups. They allow us to define quotient groups, which are groups formed by partitioning a group into cosets of a normal subgroup. Normal subgroups also help us understand the structure of a group and its subgroups.

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