Infinite dihedral group

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
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 $D_\infty$, is defined by the following presentation:

$D_\infty := \langle a,x \mid x^2 = e, xax = a^{-1} \rangle$.

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

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 $\langle a \rangle$ (isomorphic to the [group of integers]])and $\langle x \rangle$ (a group of order two).
nilpotency class	--	--
Fitting length	2	The Fitting subgroup is the group of integers $\langle a \rangle$.
Frattini length	1	The Frattini subgroup is trivial, because the maximal subgroups include $\langle a \rangle$ and subgroups of the form $\langle a^p, x \rangle$ for $p$ a prime number. The intersection of all these is trivial.
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 $\langle a \rangle$ or has a subgroup of index two inside $\langle a \rangle$. Therefore, every subgroup is either cyclic or dihedral, and thus every subgroup is 2-generated.

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