On 12/07/94 at 20:46:00 Martin Schoenert said:
The Cayley graph Gamma for a group G generated by a certain system of
generators < g_1, g_2, ... > is defined as follows.
>The vertices of Gamma correspond to the elements of G. From vertex v_1 >draw an edge to v_2 labelled with g_i, if and only if v_1 g_i = v_2. >Also draw an edge from v_2 to v_2 labelled g_i^-1 (or g_i'). v_1
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. 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?
Is the meaning universal, or does it depend on the author and the
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
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.
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