Infinite dihedral group

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VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

The infinite dihedral group, denoted $D_{\mathrm{\infty }}$, is defined by the following presentation:

$D_{\mathrm{\infty }}:=⟨a,x\mid x^2=e,xax=a^{-1}⟩$.

Here $e$ 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

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 $x$ is proper and self-normalizing.
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