Hi Joachim,
To answer your question the following operations can be implemented:
A)Under the group theory module:
1) an algorithm to check if a set is a group(abelian also)... if True what
is the Identity element,inverse of each element etc
eg. >>> G.isgroup()
>>> G.isabelian()
2) an algorithm to check if a subset of a group is a subgroup
eg >>> A.issubgroup(G)
if True then if A is a normal subgroup of G
I can also implement a method to create left and right cosets of a subgroup.
3) If a ∈ G, what is the order of a... is 'a' a generator for G eg. >>>
a.order(G)
4) if H and K are two subgroups the define HK... check if HK is a subgroup
of G >>> HK.issubgroup(G)
5) if H is a subgroup of G then define the quotient group G/H as the set of
all right cosets
6) Homomorphism: If G and H are two groups and if Φ is a mapping between
them then is Φ a homomorphism... if True what is the Kernel of Φ
7) Is Φ a 1-1 map(isomorphic)
I think some work on permutation groups has already been done... I could
implement whatever is not there right now
Also all the above operations can be implemented for a Ring
B)Under vector spaces:
1) an algo to check if a set V is a vector space over a field F
2) if Φ is mapping between two vactor spaces then is it a homomorphism and
1-1
3) Find the basis of V and hence find dimension(V). Similarly finding
dimension of a vector subspace
4) Then if B is a basis I can implement the Gram Shmidt process as
Krestanov said to get an orthonormal basis for V
These are few of the operations that I have though of right now... I am
sure many more can be implemented which can be added to the above list. I
am trying to figure out what all I can implement from field theory. Please
let me know if u think of anything more
Thnx,
Gaurav
On Mon, Mar 19, 2012 at 1:29 AM, Joachim Durchholz <[email protected]> wrote:
> Am 18.03.2012 20:17, schrieb [email protected]:
>
> I might be wrong, however the way I understand the question by Joachim
>> is rather what useful functionality those objects would bring?
>>
>
> That, and the examples that you mentioned, were what I was after.
>
>
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