Fundamental group of Klein bottle

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
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 fundamental group of Klein bottle is defined in the following equivalent ways:

1. It is the fundamental group of the Klein bottle.
2. It is the Baumslag-Solitar group $BS(1,-1)$.
3. It is given by the presentation:

$\langle a,b\mid bab^{-1}=a^{-1}\rangle$

Note that the group admits the infinite dihedral group as a quotient group.

Group properties

Property	Satisfied?	Explanation
finitely generated group	Yes	See presentation
finitely presented group	Yes	See presentation
one-relator group	Yes	See presentation
solvable group	Yes
metacyclic group	Yes
polycyclic group	Yes
supersolvable group	Yes
Noetherian group	Yes	Follows from being polycyclic
metabelian group	Yes
solvable group	Yes
residually nilpotent group	Yes