3
$\begingroup$

I am studying groups from I.N Herstein and he talks about the notion of homomorphism in the following way:

By this (homomorphism) one means a mapping from one algebraic system to a like algebraic system which preserves structure. We make this precise, for groups, in the next definition.

Definition: A mapping from group G into a group G' is a homomorphism if for all a,b $\in$ G, $\phi(ab)=\phi(a)\phi(b)$

There are two things I note here:

  1. He says "for groups etc etc" which I think implies that the above definition of a homomorphism is only valid for some (like groups) and not all structures while on the other hand the definition of homomorphism being a mapping that preserves structure is one which is general. Is this inference correct?

  2. The above brings me to my follow-up question - What exactly does it mean to preserve the structure and how for groups the above definition encodes that?

When I think of "a mapping that preserves structure" something like the following comes to mind:

A mapping that takes the set of equally spaced points on the X-axis to the equally spaced points on the Y-Axis.

or

A mapping that takes a set of points forming a figure (let's say a rectangle) on a plane to a set of points that form a rectangle at a different position on the plane.

As one can see my notion of a mapping that preserves structure is a very limited one. How do I understand it in the context of groups, the above definition and in a more general way so that if I read a book talking about homomorphism of some other structure besides groups I would know how it preserves structure.

$\endgroup$
1
  • $\begingroup$ As for your follow-up question, the notion of "having the same structure" can be made precise for isomorphic groups (see e.g. here:math.stackexchange.com/a/3929910/1007416 ), but the first homomorphism theorem tells you how to interpret it for "homomorphic groups". $\endgroup$
    – user1007416
    Commented Dec 30, 2021 at 5:02

2 Answers 2

3
$\begingroup$

Thoughtful question.

The abstraction that the idea of a group captures is that of symmetry. So for any set with some kind of "structure" the set of all maps from that set to itself that preserve the structure will form a group.

In the example you pose about maps of the plane that preserve rectangles you are implicitly thinking about the group of all the rigid motions of the plane. The structure that's preserved is the distances between points. The isomorphisms of the Euclidean plane are the rigid motions.

In your question, groups enter in two ways. For group morphisms (iso-, auto-, homo-) the underlying set with a structure is itself a group.

If you think about the points with integer coordinates in the number line, the group of symmetries is the integers themselves - you can translate in either direction by any integer amount.

Homomorphisms of groups (or of any other structure) capture the idea that you can "preserve" just parts of the structure. For example, if all you care about is whether an integer on the number line is an odd or even distance from $0$ there are only two symmetries that matter: you slide by an odd or an even distance. Those two symmetries make a group of order $2$: either you slide by an even or an odd distance. Then you have a homomorphism from the integers to the group of order $2$.

Related: What is an Homomorphism/Isomorphism "Saying"?

$\endgroup$
1
  • $\begingroup$ Nice answer. I though think that some ats could not be answered."Homomorphisms of groups (or of any other structure) capture the idea that you can "preserve" just parts of the structure" this mathematically is captured by the equation I mentioned in my answer. Now Vercassivelaunos mentions something more concretely as that equation simply means preserving equations. My question more specifically would why does the group have such a definiton of homomorphism? How does one come about it? How does tge fact that every group represents a symmetry action is related to that mathematical defintion ? $\endgroup$
    – Lost
    Commented Dec 30, 2021 at 10:36
1
$\begingroup$

In addition to Ethan Bolker's answer, you can think of homomorphism as maps which preserve true equations. Let's say $\varphi:G\to G'$ is a group homomorphism which maps elements $a\in G$ to elements $a'\in G'$ like $a':=\varphi(a)$. Then if, for instance, $ab=c$, it follows that $a'b'=c'$. This holds for any equation which can be built out of a finite number of operations allowed in groups (that is multiplication and inversion). For instance, if the equation $ab^{-1}a^{-1}=c(ba)^{-1}$ is true, then $a'b'^{-1}a'^{-1}=c'(b'a')^{-1}$ is also true.

This generalizes to homomorphisms of other algebraic structures. In vector spaces, homomorphisms preserve equations involving linear combinations of vectors. In rings, homomorphisms preserve equations involving polynomials. In metric spaces, they're called isometries instead, and the preserve equations involving the metric. And so on.

$\endgroup$
5
  • $\begingroup$ "In vector spaces, homomorphisms preserve equations involving linear combinations of vectors. In rings, homomorphisms preserve equations involving polynomials." Is this how we define homomorphism or is this a natural definiton? $\endgroup$
    – Lost
    Commented Dec 30, 2021 at 10:37
  • $\begingroup$ For instance, why defining homomorphism of vectors as equations preserving polynomials don't make sense? I guess it is the characterstic structure that decides the how homomorphism would look mathematically (preserving linear combination or polynomial equations) but I can't clearly understand it. $\endgroup$
    – Lost
    Commented Dec 30, 2021 at 10:39
  • $\begingroup$ It's always about equations which we can construct using only the given structure. Vector spaces come equipped with only vector addition and scalar multiplication, so homomorphisms only preserve those equations which contain these operations and no others. That leaves us with only linear combinations. Vector spaces don't come with a notion of polynomial equations in the first place. Rings do, since polynomials are built from ring operations only (addition and multiplication). So ring homomorphisms do preserve polynomial equations. $\endgroup$ Commented Dec 30, 2021 at 11:18
  • $\begingroup$ "Preserving structure" is much more than "preserving equations". In topology the continuous functions interact appropriately with topological notions. In set theory all functions matter. Bijections are the "isomorphisms" - they preserve cardinality, which is the only structure. $\endgroup$ Commented Dec 30, 2021 at 16:02
  • $\begingroup$ @EthanBolker Sure, structure preserving is more than equation preserving. But the structure preserving maps which have "homomorphism" in their name usually preserve structure in a way that can be expressed by equations. It was on purpose that I didn't mention continuous, measurable, differentiable, etc. maps. Though I admit that mentioning isometries in this context could be slightly misleading. $\endgroup$ Commented Dec 30, 2021 at 17:44

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .