Central series quotient of wreath product of groups of order p

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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

  1. The abelianization of a wreath product of groups of order p is an elementary abelian group of prime-square order, i.e., a direct product of two copies of the cyclic group of order p.
  2. The quotient of the group by its commutator with its commutator subgroup is a non-abelian group of order p^3. For odd p, it is isomorphic to prime-cube order group:U3p, the unique non-abelian p-group of order p^3 and exponent p. For p = 2, it is isomorphic to dihedral group:D8.
  3. The inner automorphism group of wreath product of groups of order p is a group of order p^p. It is a regular p-group. For odd p, it is also a maximal class group of exponent p, and is not an absolutely regular p-group.

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