# Introduction

A reflection is a diagonalisable linear transformation of finite order whose space of fixed points is a hyperplane. A reflection group is a finite dimensional linear group over a field $F$, which is generated by a finite number of reflections.

There are no restrictions on the field $F$ and there is no requirement for a reflection to be a transformation of order two. However, if $F$ is a real field, every reflection does have order two and there is a much richer theory. In particular, every Coxeter group is a real reflection group (see Chapter [Coxeter Groups](../CoxeterGroups/index-coxeter-groups.md#chapgrpcox)).

The books [[Lehrer and Taylor, 2009](../../references.md#cite-lehrertaylor)], [[Broué, 2010](../../references.md#cite-broue)] or [[Kane, 2001](../../references.md#cite-kanerefl)] are useful references for complex reflection groups. Standard references for the theory of real reflection groups include [[Bourbaki, 1968](../../references.md#cite-bourbakilie)] (Chapters 4, 5, 6) and [[Humphreys, 1990](../../references.md#cite-humphreysrefl)].