Re: Efficient way to store an isomorphism?

OK, I will explain it a little.
I suppose that:
- You know group theory, in particular the theory of finite groups,
- you actually want to store a morphism (or isomorphism, which is a
bijective morphism) in an efficient way.

In finite groups, a morphism h:A->B is determined by its evaluations in
the elements of A. In fact, the morphism is determined by its
evaluations in some elements of the group (maybe all, but this rarely
happen). So, the idea is to store the values of the morphism in
certains elements of A (you have to choose these elements, I think that
a generator for each subgroup can be a good choice). Then, when asked
for a certain particular value you can calculate it on the fly using
the morphism property: h(a)+h(b)=h(a*b), where + and * are arbitrary
operations. Note that you must know the decomposition of a*b, ie, you
must know a and b to calculate it.

If it is not clear, I encourage you to ask to your local mathematics
geek. I think this is a good method, but you need to understand the
foundations.

My idea actually came (but this is a little different, because vector
spaces have an scalar product and they have bases) from the
specification of linear functions in vectorial spaces. Even if a vector
space has infinite elements, the linear functions are determined by a
finite matrix, which dimensions depends on the dimensions of the
vectorial spaces involved.


I hope this will be helpful. If not, ask your local mathematician. I
acknowledge that this is a little criptic if you don't know the
subject. Don't waste time and ask someone to read it in that case.

Good luck

.

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