Amalgamated free product of Z and Z over 2Z

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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

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.