# Infinite dihedral group

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## 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.

## 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