Infinite dihedral group
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Definition
The infinite dihedral group, denoted , is defined by the following presentation:
.
Here 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
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order ((number of elements, equivalently, cardinality or size of underlying set) | Infinite (countable) | Not a finite group. |
exponent | Infinite | Not a periodic group. |
derived length | 2 | The group of integers is a subgroup of index two (explicitly, the infinite dihedral group is a semidirect product of ![]() ![]() |
nilpotency class | -- | -- |
Fitting length | 2 | The Fitting subgroup is the group of integers ![]() |
Frattini length | 1 | The Frattini subgroup is trivial, because the maximal subgroups include ![]() ![]() ![]() |
subgroup rank of a group | 2 | The whole group is 2-generated, so the subgroup rank is at least 2. Any subgroup is either inside ![]() ![]() |
Group properties
Property | Satisfied? | Explanation |
---|---|---|
abelian group | No | |
centerless group | Yes | |
nilpotent group | No | It is a nontrivial centerless group. |
hypercentral group | No | |
group satisfying normalizer condition | No | The subgroup generated by ![]() |
residually nilpotent group | Yes | |
hypocentral group | Yes | follows from being residually nilpotent |
metacyclic group | Yes | |
supersolvable group | Yes | |
polycyclic group | Yes | |
metabelian group | Yes | |
solvable group | Yes | |
finite group | No | |
2-generated group | Yes | |
finitely generated group | Yes | |
residually finite group | Yes | |
Hopfian group | Yes | finitely generated and residually finite implies Hopfian |
group with finitely many homomorphisms to any finite group | Yes | finitely generated implies finitely many homomorphisms to any finite group |
group in which every subgroup of finite index has finitely many automorphic subgroups | Yes | finitely generated implies every subgroup of finite index has finitely many automorphic subgroups |