# Central series quotient of wreath product of groups of order p

From Groupprops

## Definition

A **central series quotient of wreath product of groups of order p** is a quotient of a wreath product of groups of order p by a member of its lower central series (which in this case equals the upper central series, because the group is a maximal class group).

## Particular cases

- The abelianization of a wreath product of groups of order is an elementary abelian group of prime-square order, i.e., a direct product of two copies of the cyclic group of order .
- The quotient of the group by its commutator with its commutator subgroup is a non-abelian group of order . For odd , it is isomorphic to prime-cube order group:U3p, the unique non-abelian -group of order and exponent . For , it is isomorphic to dihedral group:D8.
- The inner automorphism group of wreath product of groups of order p is a group of order . It is a regular p-group. For odd , it is also a maximal class group of exponent , and is not an absolutely regular p-group.

Note that for , (1) and (3) coincide, and (2) coincides with the whole group. For (2) and (3) coincide.