Why only normal subgroup is used to obtain group quotient

In summary, the conversation discusses the concept of cosets and quotient groups in group theory. It is explained that for a general group and subgroup, the set of left-cosets and right-cosets will not form groups unless the subgroup is normal. This is because the product defined on the cosets will only work if the left and right cosets are the same. However, even for non-normal subgroups, the set of cosets still has a structure of a homogeneous space, which has applications in group theory and geometry.
  • #1
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Hello!

As far as I know any subgroup can, in principle, be used to divide group into bundle of cosets. Then any group element belongs to one of the cosets (or to the subgroup itself). And such division still is not qualified as a quotient.

Yes, the bundle of cosets in this case will be different for actions from the right and from the left (although, their number will be the same). But why is that so crucial? We have our division without intersections anyway, do we?

Is there any special name for such «one-sided (pseudo)quotients». Are there any uses for them?
 
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  • #2
For a general group G and a general (non normal) subgroup H, the set of left-cosets G/H, respectively the set of right-cosets H\G won't be groups, you need a normal subgroup (i.e. G/H = H\G) for that.
 
  • #3
Not quite sure I understand how cosets can be groups themselves. They lack identity element, since it is already included in the generating subgroup (normal or non-normal).
 
  • #4
You want the set of cosets to be a group (ie. the quotient group).
Say, you have a group ##G## and a subgroup ##H##. So if we want to define a product on ##G/H## (where the elements are now left-cosets), we do it like ##(aH)\cdot(bH) = (ab)H##. However, this will only make sense iff the left-cosets ##aH## are the same as the right-cosets ##Ha##, or ##aHa^{-1}=H##. Note however that for a non normal H we have ##aHa^{-1} \neq H##. This means in particular that ##(aH)\cdot(a^{-1}H)\neq (a a^{-1})H = H##. For a normal H this does work, and the rest of the group axioms are satisfied by ##G/H## with the defined product as well. Only then can we call ##G/H## a quotient group, otherwise its just a set of left-cosets.
 
  • #5
Thanks a lot for such a detailed explanation. It answers my question fully.
 
  • #6
In the case of nonnormal ##H##, the quotient still has a nice structure of a homogeneous space. That is, there is the obvious group action ##G## on ##G/H## by putting ##g\cdot kH = gkH##. This is an important action not only in group theory, but also in geometry since a lot of nice geometries arise as homogeneous spaces.
 

1. Why is it necessary to use a normal subgroup when obtaining a group quotient?

The use of a normal subgroup is necessary because it allows for a well-defined quotient group to be formed. If a subgroup is not normal, then the cosets of that subgroup may not form a group under the quotient operation.

2. What makes a subgroup "normal"?

A subgroup is considered normal if its left and right cosets are equal. In other words, the subgroup is closed under conjugation by any element in the original group.

3. Can a non-normal subgroup be used to obtain a group quotient?

No, a non-normal subgroup cannot be used to obtain a group quotient. This is because the cosets of a non-normal subgroup may not form a group under the quotient operation, leading to an undefined quotient group.

4. Are there any advantages to using a normal subgroup when obtaining a group quotient?

Yes, there are several advantages to using a normal subgroup. One advantage is that it guarantees a well-defined quotient group. Additionally, normal subgroups have a special relationship with the original group, making it easier to analyze and understand the structure of the quotient group.

5. Can a normal subgroup be used to obtain a group quotient in any group?

No, not all groups have a normal subgroup that can be used to obtain a group quotient. Only certain groups, such as abelian groups and some non-abelian groups, have normal subgroups that can be used in this way. It ultimately depends on the specific group and its properties.

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