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

From Groupprops

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

## Facts used

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