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Making a collection of sets as a group

  1. Oct 12, 2008 #1
    Any example of making a collection of sets as a group?

    Let's say, we have a collection of sets, called H. Each element of H is a set, and it works as a group element. So we have a group H whose elements correspond to sets.
    The group can be constructed easily for some trivial cases.

    For instance, H = {...,{-3}, {-2}, {-1}, {0}, {1}, {2}, {3},,,}, where H consists of single-element sets and each {x} corresponds to an integer x∈Z.

    addition : {a} + {b} = {c} (a,b,c ∈ Z}
    identity : {0}
    inverse of {a} = {-a}

    When I tried to make a collection of different size of sets, I couldn't figure out how to define multiplication (or addition), identity and inverse on sets for group operations.

    Any advice or example?

    Thanks in advance.
     
    Last edited: Oct 12, 2008
  2. jcsd
  3. Oct 12, 2008 #2

    morphism

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    Try using symmetric difference. You may have to designate an element of H to act like the empty set.
     
  4. Oct 13, 2008 #3
    Thanks a lot. It is a good example for me.
    Now I have a further question.

    Is there any case for above H to be both topology and group?
    For example, we have a collection of sets, called H. Let H be a topology on an arbitrary set X and suppose we define a group operation on H such that each open set except H itself (since a group does not have an element of itself) corresponds to each group element.

    Is it possible to define H like above or similar way such that H to be both a topology and group? If possible, any example for this?
     
  5. Oct 13, 2008 #4

    morphism

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    Well, yeah, I guess H can be a topology. But of course it doesn't have to be.

    As an example, take any (nonempty) set X. Let H be the set of all subsets of X. H is a topology on X (the discrete topology). And we can turn it into a group.
     
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