Every group of prime power order is a subgroup of a group of unipotent upper-triangular matrices

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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 group of order m = p^n for some prime p. Then, G is isomorphic to a subgroup of U_m(p), where U_m(p) is the group of upper-triangular m \times m matrices with 1s on the diagonal, over the prime field F_p.

Related facts

Facts used

Proof

Given: A group G of order m = p^n for some natural number n.

To prove: G can be embedded as a subgroup of U_m(p), the group of upper triangular unipotent m \times m matrices over the field of p elements.

Proof: By Cayley's theorem (fact (1)), G is a subgroup of the symmetric group on m elements. This, in turn, is a subgroup of the general linear group GL_m(p), under the embedding that sends each permutation to its corresponding permutation matrix. Thus, G embeds as a p-subgroup of GL_m(p).

Now, the group U_m(p) is a p-Sylow subgroup of GL_m(p), so by fact (2), some conjugate of G lies inside U_m(p). Since this conjugate subgroup is in particular isomorphic to G, we obtain an embedding of G as a subgroup of U_m(p).