Incompressible Groups

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.

Name thesis:degree_name

Master of Science in Mathematics

Level thesis:degree_level

Masters

Discipline thesis:degree_discipline

Mathematics

Year

2002

Author dc:creator

• Aberle, Andrew

Contributors dc:contributor

• Les Reid

• Mathematics

Statement dc:rights

• © Andrew Aberle

Repository record dc:identifier

https://bearworks.missouristate.edu/theses/859

OAI identifier oai:identifier

oai:bearworks.missouristate.edu:theses-1860