1. The group theory commands are described in
http://sage.scipy.org/sage/doc/html/ref/node156.html (the reference
manual under "Groups")
http://sage.scipy.org/sage/doc/html/const/node5.html (the "How to ..."
under "Groups")
http://sage.scipy.org/sage/doc/html/tut/node4.html#SECTION004500000000000000000
(the Turorial under "Finite groups")
2. I'll be adding commands this weekend (for a completely unrelated reason)
and can add an "is_solvable" command. However, there is a "derived_series"
command and using it you can determine if the group is solvable (The
group is solvable iff the derived series terminates in the trivial
group {1} after a finite number of steps.)
3. It does not check if an array defines a group operation./ /The
easiest way to do this in SAGE is
to create n permutations matrices from your multiplication table p1,
..., pn. (The ij-th entry
of pk is 1 iff the k-th element of your "group" occurs in the ij-th
entry of the table.) The SAGE
can construct the matrix group generated by p1, ..., pn. If this group
is also order n then you are
in luck. Otherwise not. Such a function would be easy enough to write in
SAGE but I don't know how
much use it will get. It would be useful for teaching about groups in an
undergrad algebra or
group theory course. Do you think it should be added?
4. SAGE will identify subgroups of a permutation group (I think this is
what you mean).
5. I'm not such what you mean by "convert ... to an array". Do you mean
convert from
disjoint cycle notation in SymmetricGroup(n) to 2xn array notation for
the group elements?
No SAGE doesn't currently do this, but again, that would be easy to do. /
/6. Yes, SAGE handles groups with several generators./
/Thanks for trying out SAGE. Please let me know what you think!/
++++++++++++++++++++++++++++++++++++++++++++
/Richard Boardman wrote:
> I am working through your book and have also downloaded the SAGE package but
> I can't find a general description of what it will do in relation to groups.
> If I define a set of n objects and an array of n*n elements defining an
> operation, will it check if this forms a group, will it check if the group
> is arbelian? or solvable? will it identify subgroups, will it convert an
> operation defined in cyclic terms to an array?, will it help where there are
> multiple generators?
>
> many thanks for your help
>
> Dick Boardman
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