Amalgamated free product of Z and Z over 2Z

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Definition

Definition as an amalgamated free product

The group is defined as the amalgamated free product \mathbb{Z} *_{2\mathbb{Z}} \mathbb{Z}, i.e., we take two copies of the group of integers, take their free product, and then take the quotient group by the identification of the subgroup 2\mathbb{Z}.

Definition by presentation

The group can be defined by the presentation:

\langle x,y \mid x^2 = y^2 \rangle

Group properties

Property Meaning Satisfied? Explanation
abelian group any two elements commute No x,y do not commute
nilpotent group upper central series reaches the whole group in finitely many steps No The quotient group by the center \langle x^2 \rangle is isomorphic to the infinite dihedral group, which is centerless.
solvable group has a normal series where all the quotients are abelian Yes The quotient by the center is infinite dihedral group, which is a metacyclic group.