# Amalgamated free product of Z and Z over 2Z

From Groupprops

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Contents

## Definition

### Definition as an amalgamated free product

The group is defined as the amalgamated free product , 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 .

### Definition by presentation

The group can be defined by the presentation:

## Group properties

Property | Meaning | Satisfied? | Explanation |
---|---|---|---|

abelian group | any two elements commute | No | do not commute |

nilpotent group | upper central series reaches the whole group in finitely many steps | No | The quotient group by the center 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. |