A generalization of the quotient group: an exhaustive analysis of all possible cases
We know that if is a group and is a normal subgroup of , then is a quotient group.
Today, we shall try and explore some fundamental questions that have plagued my understanding of Algebra for a long long time.
What if is just a subset of , and not necessarily a subgroup? Does mean anything then? Take and . Then consists of for all . For two different , we have . Hence, there exists a bijection .
Can display properties of a group? Sure. Define a new operation on . We have a two-sided identity, a two-sided inverse, associativity, and algebraic closure. We have a group!!
What happens when becomes a subgroup? For any , we see that . Imagine that for each equivalence class, we fix a representation; i.e. for the equivalence class , we choose the element for all arithmetic operations, without the need to call any other element for any further operations. Let and be the representative elements of two equivalence groups in . What about ? Can we prove that we’ve chosen to be the representative element of some equivalence class? Probably not.
Now let be a normal subgroup. Here . So we have , where . However, we can’t be sure of choosing this element as the representative of some equivalence class either.
Instead of choosing representative elements, let us try a different approach. Let us consider whole cosets at once for . Choosing representative elements becomes irrelevant here. For group operations, we’d have to calculate ; i.e. add each element of to each and every element of .
If is normal, then we have , which surely is an element of . Checking for other group properties, we soon see that is a group.
What if is not normal? Algebraic closure may be violated. Check out this link: http://math.stackexchange.com/questions/14282/why-do-we-define-quotient-groups-for-normal-subgroups-only
If is a normal subgroup, then . Note that this isn’t a necessary condition for to be a group As long as for some , we would still have algebraic closure. This is just an additional luxury that provides for notational convenience.
Conclusion: It is possible for to be a group if is not a subgroup at all. The group operation would be . Remember that this would be a result of using the operation on the elements of and . The operation on would just have to be defined this way. Choosing representatives doesn’t even come up here as there is a different for every .
When is a subgroup, there is one coset for multiple elements in . Hence, choosing representatives comes up. But this proves to be problematic whilst trying to satisfy algebraic closure. Hence, we deal with adding whole cosets to each other. Choosing representatives becomes irrelevant here. For the purpose of satisfying algebraic closure, we find that if is a normal subgroup, is a subgroup in every instance.