Generalized dihedral group for additive group of 2-adic integers

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Definition

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

• 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.

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