{"id":{"repo_id":"mo-state","oai_identifier":"oai:bearworks.missouristate.edu:theses-1860"},"canonical_url":"https://search.dev.ndltd.org/etd/mo-state/oai:bearworks.missouristate.edu:theses-1860","repository":{"repo_id":"mo-state","name":"Missouri State University","base_url":"https://bearworks.missouristate.edu/do/oai/"},"display":{"title":"Incompressible Groups","abstract":"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.","abstract_html":"In group theory, Cayley&#x27;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 &lt; 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&lt;4, hence, Z₄ is incompressible. In contrast, S₃ has order 6 so by Cayley&#x27;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.","abstract_has_math":false,"creators":["Aberle, Andrew"],"institution":null,"degree_name":"Master of Science in Mathematics","degree_level":"Masters","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Les Reid"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2002,"date_issued":"2002-07-01T07:00:00Z","date_published":"2002-07-01T07:00:00Z","updated_at":"2026-07-24T03:15:54Z","subjects":["Mathematics"],"languages":[],"rights":["© Andrew Aberle"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://bearworks.missouristate.edu/theses/859","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Les Reid"]},{"key":"dc:creator","label":"Author","values":["Aberle, Andrew"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science in Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["© Andrew Aberle"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://bearworks.missouristate.edu/theses/859"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:title","label":"Title","values":["Incompressible Groups"]}]}],"canonical_facts":{"dc:contributor":["Les Reid"],"dc:creator":["Aberle, Andrew"],"dc:description.abstract":["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."],"dc:identifier":["https://bearworks.missouristate.edu/theses/859"],"dc:rights":["© Andrew Aberle"],"dc:subject":["Mathematics"],"dc:title":["Incompressible Groups"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Masters"],"thesis:degree_name":["Master of Science in Mathematics"]},"updated_at":"2026-07-24T03:15:54Z"}