In group theory, Cayley's theorem states that if G is a group of order n, then G is isomorphic to a subgroup of the symmetric group, Sn. This theorem gives rise to the question: Can we improve on the index n? To consider this, we need the following definitions: A group G, with order n, is said to be incompressible if it is not isomorphic to a subgroup of Sm where m < n, and if a group G is not incompressible then it is said to be compressible. For example, we will see that Z₄ is not isomorphic to any subgroup of Sn where n<4, hence, Z₄ is incompressible. In contrast, S₃ has order 6 so by Cayley's theorem we know that S₃ is isomorphic to a subgroup of S₆. Although this is true, S₃ is clearly isomorphic to a subgroup of S₃, so it is compressible. The purpose of this paper is to prove a necessary and sufficient criterion to determine what groups of order n are incompressible.
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