# Can We Identify Quotient Groups as Subgroups of the Original Group?

• Hello Kitty
In summary, the conversation discusses identifying a quotient group G/M with a subgroup of G/N. The approach of finding a homomorphism from G to G/N with kernel M is considered, but no counterexample was found. The conversation also mentions a more specialized result and discusses the need for automorphism groups to select from in order to find a group where G/M operates differently on M than G/N does on N.
Hello Kitty
Let G be a group and let $$N\trianglelefteq G$$, $$M\trianglelefteq G$$ be such that $$N \le M$$. I would like to know if, in general, we can identify $$G/M$$ with a subgroup of $$G/N$$.

Of course the obvious way to proceed is to look for a homomorphism from $$G$$ to $$G/N$$ whose kernel is M, but I can't think of one.

What I actually want to show is a more specialized result (namely the case when finite $$G/N$$ is the nilpotent quotient of $$G$$ and $$G/M$$ is a maximal p-quotient of $$G$$ for some p dividing the order of $$G/N$$) but the above is a lot cleaner and didn't yield obviously to a proof or counter-example so I thought I'd explore that first.

I would look for a counterexample, although I found none as the small groups are all "too cyclic". We have
$$M\rtimes G/M \cong G \cong N \rtimes G/N\text{ and } N \triangleleft M$$
This means we have to look for the automorphisms and need a group, where ##G/M## operates differently on ##M## than ##G/N## does on ##N##. Therefore we need automorphism groups which have a few elements to select from.

## 1. What are subgroups of quotient groups?

Subgroups of quotient groups are subgroups of a larger group formed by taking the original group and dividing out by a normal subgroup. They can be thought of as a smaller version of the original group that retains certain properties.

## 2. How are subgroups of quotient groups related to normal subgroups?

Subgroups of quotient groups are formed by taking the original group and dividing out by a normal subgroup. This means that the normal subgroup is a subset of the subgroup of the quotient group. Additionally, the quotient group is isomorphic to the original group divided by the normal subgroup.

## 3. What is the significance of subgroups of quotient groups?

Subgroups of quotient groups have practical applications in group theory and abstract algebra. They help us understand the structure of groups and can be used to prove theorems and solve problems related to groups and their properties.

## 4. How do you determine if a subgroup of a quotient group is normal?

A subgroup of a quotient group is normal if and only if it is the kernel of a homomorphism from the original group to another group. In other words, if every element in the subgroup is mapped to the identity element of the other group, then it is normal.

## 5. Can subgroups of quotient groups have subgroups?

Yes, subgroups of quotient groups can have subgroups. In fact, every subgroup of a quotient group is also a subgroup of the original group, and can have its own subgroups as well. This is because subgroups of quotient groups retain the same structure as the original group, just on a smaller scale.

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