A Coxeter system is a pair (W, S) where W is a group and where
S is a set of involutions in W such that W has a presentation of the form
Here m(s, t) denotes the order of st in W
and in the presentation for W, (s, t) ranges
over all pairs in S × S such that m(s, t) ≠ ∞.
We further require the set S to be finite.
W is a Coxeter group and S is a fundamental set of generators for W.
Obviously, if S is a fundamental set of generators, then so is
wSw−1, for any w∈W. Our main result is that,
under certain circumstances, this is the only way in
which two fundamental sets of generators can differ. In Section 3, we will prove
the following result as Theorem 3.1.