Isomorphisms and homomorphisms

i was just wondering if someone (matt) could give me a better idea of what the difference is between the two...thanks
 

AKG

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The difference is incredibly basic, it should appear in any textbook, and you could have easily looked it up on the web. A homomorphism is a map f : G -> H between groups such that f(gg') = f(g)f(g') for any g, g' in G. An isomorphism is a bijective homomorphism.
 
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yeah an isomorphism is a homomorphism which is also 1-1 & onto. so there are 3 things you need to check to show that a function between two algebraic structures are isomorphic under a function f:
i) for every x, y in the domain, f(xy) = f(x)f(y)
ii) f(x)=y for some y in the target space
iii) if f(x)=f(y) in the target space then x=y in the domain
 

AKG

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If you are speaking of algebraic structures in general, then you may have to check more than those three things. The definitions I gave were for group homomorphism and isomorphisms. For a ring homomorphism f : R -> S, it must hold that:

f(r + r') = f(r) + f(r')
f(rr') = f(r)f(r')

for any r, r' in R. Then, as it was for groups, a ring isomorphism is a bijective ring homomorphism. Although I've never seen this terminology used, a vector space homomorphism T : V(F) -> W(F) (i.e. V and W are both vector spaces over the field F) is a map such that:

T(av) = aT(v) for a in F, v in V
T(v + v') = T(v) + T(v') for v, v' in V

Then a vector space isomorphism is a bijective vector space homomorphism. Hopefully you recognize the above as the conditions for T to be a linear transformation. So a linear transformation of vector spaces is precisely a vector space homomorphism. So a v.s. isomorphism is a bijective homomorphism, and thus a bijective linear transformation, as you should already have known.

The general idea of a homomorphism is that it "respects" algebraic structure, where the algebraic structure of some thing is captured by its operations. The operations for a vector space are scalar multiplication and vector addition, and you can see how our v.s. homomorphism "respect" those operations. For groups, the operation is just multiplication, and for rings, the operations are addition and multiplication, and in both these cases, you can see how the homomorphisms respect (or "preserve") these operations. As far as I know, regardless of what type of algebraic structure we're talking about, an isomorphism will always be a bijective homomorphism.
 

Hurkyl

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yeah an isomorphism is a homomorphism which is also 1-1 & onto.
Not quite -- an isomorphism is a homomorphism that has an inverse (that is also a homomorphism).

In some cases, this means exactly what you said. In other cases, it does not. For instance, when discussing topological spaces, the "homomorphisms" are the continuous maps, and "isomorphisms" are the homeomorphic maps. There exist bijective homomorphisms whose inverse is not continuous.
 
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Hurkyl said:
Not quite -- an isomorphism is a homomorphism that has an inverse (that is also a homomorphism).
In some cases, this means exactly what you said. In other cases, it does not. For instance, when discussing topological spaces, the "homomorphisms" are the continuous maps, and "isomorphisms" are the homeomorphic maps. There exist bijective homomorphisms whose inverse is not continuous.
i guess those are the analogues in topology... never heard of them explained that way before. i was referring to groups & rings though & yeah i forgot the other operation in the ring. argh :eek:
 

Hurkyl

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Sometimes, you might want to consider things other than plain vanilla groups and rings!

For example, you might want to consider topological rings: rings with a topology whose ring operations are continuous.

The appropriate definition of homomorphism for such things is that a function is a homomorphism of topological rings iff it is a continuous homomorphism of rings.

And then, we can find bijective homomorphisms of topological rings that are not isomorphisms.
 
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see http://planetmath.org/encyclopedia/Isomorphism2.html [Broken] for excellent discussion on topic.
 
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mathwonk

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a homomorphism is a way of comparing two algebraic objects. when the comparison shows they are the same it is called an isomorphism, since then it has an inverse. i.e. isomorphism equals homomorphism with inverse. (sadly for us, matt is taking a hiatus from the forum.)
 
Not quite -- an isomorphism is a homomorphism that has an inverse (that is also a homomorphism).

In some cases, this means exactly what you said. In other cases, it does not. For instance, when discussing topological spaces, the "homomorphisms" are the continuous maps, and "isomorphisms" are the homeomorphic maps. There exist bijective homomorphisms whose inverse is not continuous.
I am sorry to jump in by a very elementary question: is there a way to explain with non-mathematical examples the concept of iso- and homo- morphism? Or to downgrade the level of the mathematical explanation so as to fit brains that are slightly poor in its understanding capabilities for this domain? Thanks so much !
 

mathwonk

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the us senate and congress are sort of homomorphic to the georgia state senate and congress, but not isomorphic because the state ones are smaller.
 
Thank you very much for your kind and very clear answer about isomorphism. Now, if i understand well, homomorphic means the same structure and isomorphic means an equal structure. Is there any way to name the identity, the relation between something and itself? I mean, "homo" and "iso" morphisms hold between more than one and it is very usefull to understant what one means when saying "i have the seen the same car", could be the same brand and model with a different plate or just the very same (same plate).
 

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