Every group of prime power order is a subgroup of an iterated wreath product of groups of order p

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This article gives the statement, and possibly proof, of an embeddability theorem: a result that states that any group of a certain kind can be embedded in a group of a more restricted kind.
View a complete list of embeddability theorems

Statement

Suppose G is a Group of prime power order (?), i.e., a finite p-group for some prime p. Then, G can be embedded as a subgroup of a p-group P, where P is an iterated wreath product of cyclic groups of order p. (Note that if P is the wreath product of n such groups, P is isomorphic to the p-Sylow subgroup of the symmetric group on a set of size p^n).

Facts used

  1. Cayley's theorem
  2. Sylow implies order-dominating: The domination part of Sylow's theorem, which states that given a p-subgroup and a p-Sylow subgroup, some conjugate of the p-subgroup lies in the p-Sylow subgroup.