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Math Help - Nature of Homomorphisms

  1. #1
    Super Member Bernhard's Avatar
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    Nature of Homomorphisms

    I am a math hobbyist reading John Fraleigh's book on Abstract Algebra.

    In Section 13 on Homomorphisms Fraleigh gives the following example. { I have put r =2 in his example to make it more tangible and specific)

    Let r = 2  \in Z and let  \phi_{2} : Z \rightarrow Z be defined by  \phi_{2} (n) = 2n for all n  \in Z.

    For all m,n  \in Z, we have  \phi_{2} (m+n) = 2(m+n) = 2m + 2n = \phi_{2} (m) + \phi_{2} (n) \ so \ \phi_{2} is a homomorphism.

    Although Fraleigh expresses this as a homomorphism of  Z \rightarrow Z it seems to me the target group must be the additive group 2Z = { ..., -4, -2, 0, 2, 4, 6, ...} and not Z as implied by his definition of the homomorphism. [Note that Fraleigh defines a groups homomorphism as being between a group G and a group G' but in the example seems to be referring to a mapping between two sets, with the target group being 2Z which, obviously is a subset of Z]

    I would be grateful if someone could clarify this for me

    Bernhard
    Last edited by Bernhard; March 25th 2011 at 04:08 AM.
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  2. #2
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    there is nothing in the definition of a homomorphism that requires it to be a surjective map (onto function). surjective homomorphisms are of special importance, however, and the image of a homomorphism φ:G-->G' is usually what one is most interested in.

    remember that for a function (any function) f:A-->B, all that is required is that f(A) be a subset of B. one does not, for example, think of f:R-->R given by f(x) = x^2 as being a "non-negative real-valued function" (although it is), but simply as a "real-valued function" even though its range is not the entire co-domain.
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  3. #3
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    No, that is a perfectly good example of a homomorphism from Z to Z- a homomorphism does NOT have to map "onto" the entire set- it can map to a subset. In fact (its probably in the next chapter!) a homomorphism that maps from A to B both "one to one" (f(a)= f(b) only if a= b) and "onto" (maps to the entire set) is an "isomorphism", a very special kind of homomorphism.

    So this could be consider as a homomorphism from Z to Z or an isomorphism from Z to 2Z.
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  4. #4
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    to clarify even further, there are 4 types of homomorphisms:

    just plain homomorphisms (sometimes just called morphisms, terminology varies).
    epimorphisms, or surjective (onto) homomorphisms.
    monomorphisms, or injective (1-1) homomorphisms.
    isomorphisms, or bijective homomorphisms (both epimorphisms and monomorphisms).

    a common mistake people often make is to show a homomorphism is an isomorphism by showing it is 1-1. a monomorphism φ:G-->G' IS an isomorphism of G with φ(G), but not necessarily one of G with G'. the example you give exhibits an isomorphism of Z with 2Z (which implies, among other things, that Z must be an infinite group, as it is isomorphic to a (proper) subgroup of itself. this cannot happen with a finite group).
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