Making a collection of sets as a group

[infinityQ]

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.

 

[morphism]

Try using symmetric difference. You may have to designate an element of H to act like the empty set.

 

[infinityQ]

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?

 

[morphism]

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.