Infinite dihedral group

Definition
The infinite dihedral group, denoted $$D_\infty$$, is defined by the following presentation:

$$D_\infty := \langle a,x \mid x^2 = e, xax = a^{-1} \rangle$$.

Here $$e$$ denotes the identity element.

Equivalently, it is the generalized dihedral group corresponding to the additive group of integers.

Related groups

 * Generalized dihedral group for additive group of 2-adic integers
 * Generalized dihedral group for 2-quasicyclic group