## Review of Group Theory: Homomorphisms

**Point of post: **In this post we review basic facts about homomorphisms. Simple stuff like the image of a subgroup is a subgroup, the inverse image of a subgroup is a subgroup, etc. etc.

*Motivation*

Just as linear transformations are the structure preserving maps for vector spaces homomorphisms are the structure preserving maps for groups. They, in essence, preserve most the qualities that interest us in the theory of groups. And, if the homomorphism happens to be bijective then we get a group isomorphisms (not to be confused with a linear isomorphism) which preserve ‘all’ the qualities that interest us. So, let us begin

*Homomorphisms*

Let and be groups. Then a map is a *group homomorphism *if

for all . We denote the set of all homomorphisms by . We prove the immediate theorems

**Theorem: ***Let ,, and be groups and and be homomorphisms then:*

**Proof:**

: This is immediately clear since and so upon cancellation .

: It’s clear by induction that for all . Note though that by we have that

and thus . Thus, it follows that

for all from where the problem easily follows.

: Let then

Thus, noting that we may conclude from a previous theorem that .

: Suppose that , then

so that . Noticing then that allows us to conclude.

: This is similar to the proof that set-theoretic inverses of linear transformations are linear transformations. Namely, let . Then, since is surjective there exists such that and . Then,

and thus

from where the conclusion follows.

: This follows since if is arbitrary there exists such that

where . Then,

from where the conclusion follows.

: Let be arbitrary. Since is surjective there exists such that and . So,

since were arbitrary the conclusion follows.

: We merely note that

: This follows from noticing that for any we have that

**References:**

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

2. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print

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