# 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.