Definition 3

Let L be a hermitian space over a field A, T the space of translations of L. A displacement (resp. similitude) of L is any affine bijection u of L onto L such that the linear mapping $\varphi$ associated with u in T (Chapter II, 2nd ed., App. II, No. 4) is unitary (resp. is a similitude).

The group of translations is a normal subgroup of the affine group; it is therefore a normal subgroup of the group of similitudes and of the group of displacements. For every $a \in L$, let $G_a$ be the group of similitudes (resp. displacements) leaving a fixed; if L is identified with T by taking a as origin, $G_a$ is the group of similitudes (resp. the unitary group) of T. Every similitude (resp. displacement) u can be written, in one and only one way, in the form $u = u_1 t_1$ where $u_1 \in G_a$ and $t_1 \in T$, and also in the form $u = t_2 u_2$ where $u_2 \in G_a$ and $t_2 \in T$; moreover $u_2 = u_1$ and $t_2 = u_1 t_1 u_1^{-1}$ (Chapter II, 2nd ed., App. II, No. 4).

Let $u$ be a similitude in L, $\varphi$ the associated similitude in T. The multiplier of $\varphi$ is also called the multiplier of $u$ (No. 5). If this multiplier is denoted by $\alpha(u)$, the mapping $u \to \alpha(u)$ is a homomorphism of the group of similitudes of L into the multiplicative group of the invertible elements of the center of A; its kernel is the group of displacements, which is therefore a normal subgroup of the group of similitudes. When A is commutative and L is of finite dimension, there are, between the determinant det $u$ (equal by definition to det $\varphi$) and $\alpha(u)$, the same relations as in No. 5. The displacements $u$ such that det $u = 1$ form a normal subgroup of the group of displacements; this subgroup is of index 2 if A is a commutative field of characteristic $\neq 2$ and J is the identity.