[GAP Forum] re: some simple and obvious and MISSING items for the FAQ

shane miller clearthink at cavtel.net
Mon Oct 11 09:07:10 BST 2004

```Hi,

I installed gap without problem using gapsync. GAP is
an impressive algebra system. However, some really
obvious things are missing. Perhaps this needs mere
FAQ documentation:

1. Almost every tutorial or other documentation tells
how one can compute 43 million things (like subgroups
or cosets) on a given group but fails to:
a. tell how to perform a binary operation on the
darned elements of the group. Now, for Sn groups
this appears documented via constructions like
(1 2 3)(4 2 1) etc. But,
1. what do you if 4 or 5 Sn groups have been
constructed. Then the command (1 2 3)(4 2 1)
applies to which Sn group? How does GAP match
the binary operation request with a group
structure esp. if more than one has been defined?
Please make this general: Q8 or Z4 do not
(typically) use permutation notation.
2. how to add, say, 5+13 in some Z/Z15 (i.e.
15>5, 15>13) or Z/Z3 group?
3. How does GAP know Z/Z3 is using + (add) or
'*'? This must be implied by the groups's
generator?
2. Many classic groups like D4, Q8 can be constructed
via simple commands. But there are no elements; they're
only generators! How in the heck do you get GAP to
tell you what the actual elements are for Q8? i.e.
q8:=SmallGroup(8,4);
Elements(q8) = [f1,f2,f3] <-- not elements
In any basic algebra book Q8 is a group on 8 distinct
symbols {i,j,k,1,-1,-i,-j,-k}. How do you make GAP use
these elements? How can you get GAP to do this:
Elements(q8)={i,j,k,1,-1,-i,-j,-k}
Of course the i,j,k etc. are not special; but how can
one specify this set? And finally how would one then
ask GAP to compute, say, "-1*j" in q8?
3. Presumably, once (2) is solved, then commands like
NormalSubgroups(q8)
will print all the normal subgroups using (user
specified) elements and not darned generator expressions.
Ditto computing cosets, normalizers etc.
4. Another super feature is the GraphicSubgroupLattice.
But again, the numbered subgroups in the diagram are
not that useful. Suppose you diagram D16 and pick,
at random, subgroup #3. How in the heck do you ask
GAP to tell you the generating elements (not generator
expressions) for that subgroup (#3 here)?
5. And finally, given (4), how do you tell GAP to generate
the number #3 subgroup of D16 using the generation
information from (4)?

Shane

--
Sure it's a technical problem of fine pedigree ... But all such problems
are wrapped inside a management problem. And there's the rub.

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