Generalized dihedral group for additive group of 2-adic integers

As an abstract group
This group is defined as the defining ingredient::generalized dihedral group corresponding to the additive group of 2-adic integers, which is the additive group of $$p$$-adic integers for the prime $$p = 2$$.

In other words, it is the external semidirect product of the additive group of 2-adic integers by cyclic group:Z2, where the non-identity element of the acting group acts by the map sending an element to its negative.

As a profinite group
The group becomes a profinite group under the natural choice of profinite topology. It can also be explicitly realized as the limit of the inverse system:

$$D_8 \leftarrow D_{16} \leftarrow D_{32} \leftarrow D_{64} \leftarrow \dots$$

where each map is (essentially) quotienting out by the center. Explicitly, it can be thought of as reducing the cyclic subgroup of index two modulo a smaller power of 2 while keeping the acting part intact.

Related groups

 * Infinite dihedral group is the generalized dihedral group for the group of integers. This is a dense subgroup of the generalized dihedral group for additive group of 2-adic integers.
 * Generalized dihedral group for 2-quasicyclic group has somewhat related and dual properties to this one.