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.
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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 .
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 .