On 12/07/94 at 20:46:00 Martin Schoenert said:
So the Cayley graph depends on the group *and* on the generating system.
Simple, isn't it.
These are fine points, but they bother me anyway.
1. Suppose I write <Q>=<Q,H>. If I mean that the group <Q> is equal
to the group <Q,H>, then the equation is correct.
here you're using "<Q>" to denote the group.
If I mean that the Cayley graph of <Q> is the same as the Cayley graph of <Q,H>, then the equation is incorrect. Which is the conventional meaning?
but now you're trying to use the same symbol to denote the cayley graph.
Is the meaning universal, or does it depend on the author and the
the context should make it clear which object (group or graph) the symbols
refer to. as martin notes above, these are quite different objects.
2. I gather from your note and from things that Dan sent me that
one should not list inverses of the generators. For example,
<U,R> is sufficient and one should not write <U,U',R,R'>. But
people conventionally write <Q> which includes six processes and
their six inverses. Is this acceptable usage, or should we write
either is acceptable, whether you're referring to groups or cayley graphs.
however, for cayley digraphs (directed graphs), the two meanings are quite
different. i don't imagine we'll have anything to do with digraphs until
someone complains that they can only turn the faces of their cube clockwise,
and wants to know some short processes.
in our case, "<Q>" is preferred, since it's shorter.
As an additional comment, I have frequently written about the Q length
of a process in <U,R> or the Q+H length of a process in <U,R>. I think
we would be better served to talk about the length of a process in
<U,R> or the length of a process in <U,U2,R,R2> if the generator
notation implies a particular Cayley graph.
yes, this terminology is more precise, but the meaning was already clear.