## Review of Group Theory: Normal Subgroups and Quotient Groups (Pt. II)

**Point of post: **This post is a continuation of this one.

There are also some structure properties of normal subgroups.

**Theorem: ***Let be a group and be such that for every . Then, .*

**Proof: **To do this we merely note that for any

from where the conclusion follows.

Another clear one is

**Theorem: ***Let be a group with and . Then, .*

**Proof: **We merely note that for any

and thus from our early characterization of normality it follows that .

*Remark: *The converse of this clearly does not hold.

Lastly, we prove a commonly used theorem (which is a specific case of a more general theorem I hope I will eventually get to discuss):

**Theorem: ***Let be a group and with . Then, .*

**Proof: **It suffices to show (by our characterizations of normality) that the set of right cosets of and set for left cosets of coincide. Note though that the set of left cosets is of the form for some and the set of right cosets are of the form for some . But since both of these are partitions it follows that from where the conclusion follows.

*Quotient Groups*

We now wish to define a group structure on for normal . In particular:

**Theorem: ***Let be a group and . Then, the operation on is well-defined (in the sense that if and then ) if and only if . Moreover, if then is a group under this operation with and .*

**Proof: **Assume first that the described relation is well-defined and let and be arbitrary. Note then that by assumption

note though that and thus there exists some such that .Multiplying on the right gives and so . Since and were arbitrary it follows that for every . By our previous theorem we may conclude that .

Conversely, suppose that and suppose that are such that

we prove that

To do this we first note that clearly there exists such that and . Note though that

where . Thus, it follows that and since the left cosets of partition it follows that as desired.

The verification of the group axioms as are the stated properties about inverses and identity elements.

We call the above group structured defined on to be the *quotient group of mod *(or just mod for short). We also never denote the identity element of by but rather by just .

For finite groups there is a nice formula for the order of a quotient group. Namely:

**Theorem: ***Let be finite and . Then,*

**Proof: **This is evident from Lagrange’s Theorem.

While not particularly profound the above theorem has an interesting corollary:

**Corollary: ***Let be a finite group and . Then, for any , .*

**Proof: **By Lagrange’s theorem we have that

But, we know that this will be true if and only if .

It’s clear from definition that we have a nice surjective homomorphism from onto . Namely we constructed the operation on in such a way that the following theorem is true:

**Theorem: ***Let be a group and . Then, the map *

*is a surjective homomorphism. Moreover, .*

**Proof: **By definition we have that

and is surjective since every element of is of the form for some . To prove that we clearly note that and if then and this is true if and only if . The conclusion follows.

*Remark: *This mapping is called the *canonical projection of onto .*

**Corollary: **The converse of in our characterization of normality is now proven since every normal subgroup may be realized as the quotient of its canonical projection.

**Corollary: ***If is an abelian group and then is abelian. If is cyclic then is cyclic. In general if has a property invariant under epimorphisms (surjective homomorphisms) then has the same property.*

**References:**

1. Lang, Serge. *Undergraduate Algebra*. 3rd. ed. Springer, 2010. Print.

2. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 200

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