# Fundamental group of Klein bottle

## 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