Generalized dihedral group for additive group of 2-adic integers

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As an abstract group

This group is defined as the 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

Group properties

Abstract group properties

Property Satisfied? Explanation
p-group No The 2-adic integers have infinite order and their order is not a power of 2.
abelian group No semidirect product with nontrivial action
centerless group Yes
nilpotent group No
hypercentral group No
residually nilpotent group Yes
metabelian group Yes
solvable group Yes
finite group No
finitely generated group No
countable group No
residually finite group Yes