## Review of Group Theory: Direct Product of Groups (Pt. II)

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

The interesting thing is that these properties characterize the direct product of groups. More specifically:

**Theorem: ***Let be groups and for be such that whenever . Assume that the same universal property: if for each are such that whenever then there exists a unique such that for each . Then, .*

**Proof:** For lack of diagrams, we leave this as an exercise to the interested reader.

Another equivalent characterization is

**Theorem: ***Let be groups. Then, if and only if contains subgroups such that*

**Proof: **Suppose first that satisfies and and define the guaranteed isomorphisms

Then, consider

Evidently this an injection by and a surjection by and it’s a homomorphism since each of the commute with the when (using the same proof as in the second universal property of direct products). It follows that is an isomorphism.

Conversely, it’s fairly easy to see that the groups satisfy and for .

We end this post with a few miscellanea:

**Theorem: ***Let groups. Then:*

**Proof: **

: This follows immediately since it’s clear that if and are isomorphism then is an isomorphism.

: Define

this is evidently surjective, and a homomorphism since

Noting though that

allows us to finish the argument by the First Isomorphism Theorem.

: Suppose that . Then, we have that yet for every we have that

so that is not generated by any of its elements.

Conversely, note that evidently and thus . That said, note that since

we have that and and since we may conclude that from where the conclusion follows.

**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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