Hyperreflection groups
G
m
{G_m}
are generalizations of groups generated by reflections. A hyperreflection group is generated by hyperreflections if
dim
V
\dim V
is finite. A hyperreflection is a simple mapping
σ
\sigma
such that
det
σ
=
γ
\det \sigma = \gamma
, where
γ
m
=
1
{\gamma ^m} = 1
. If the field of scalars is commutative, the order of
σ
\sigma
is
m
m
. Our main result states that every relation between hyperreflections and their inverses is a consequence of relations of lengths 2, 4, and
m
m
. The most interesting special case occurs for
m
=
2
m = 2
. Then our result refers to relations between reflections.