Wreath product of group of integers with group of integers

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Statement

The wreath product of group of integers with group of integers is defined as the restricted external wreath product:

\mathbb{Z} \wr \mathbb{Z},

where \mathbb{Z} is the additive group of integers, and the permutation action is the regular group action. It can also be viewed as the semidirect product of the additive group of the Laurent polynomial ring over \mathbb{Z} with the multiplicative cyclic group generated by x.

Facts