Generalized dihedral group for additive group of 2-adic integers
This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]
As an abstract group
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
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.
- 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.
Abstract group properties
|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|
|residually nilpotent group||Yes|
|finitely generated group||No|
|residually finite group||Yes|