Every group of prime power order is a subgroup of a group of unipotent upper-triangular matrices
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
Suppose is a Group of prime power order (?), i.e., a group of order for some prime . Then, is isomorphic to a subgroup of , where is the group of upper-triangular matrices with s on the diagonal, over the prime field .
- Every finite group is a subgroup of a finite simple group
- Every finite group is a subgroup of a finite complete group
- Every group of prime power order is a subgroup of an iterated wreath product of groups of order p
- 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 inside the -Sylow subgroup.
Given: A group of order for some natural number .
To prove: can be embedded as a subgroup of , the group of upper triangular unipotent matrices over the field of elements.
Proof: By Cayley's theorem (fact (1)), is a subgroup of the symmetric group on elements. This, in turn, is a subgroup of the general linear group , under the embedding that sends each permutation to its corresponding permutation matrix. Thus, embeds as a -subgroup of .
Now, the group is a -Sylow subgroup of , so by fact (2), some conjugate of lies inside . Since this conjugate subgroup is in particular isomorphic to , we obtain an embedding of as a subgroup of .