When choosing k random elements from a group the kth expectation number is the expected size of the subgroup generated by those specific elements. The main purpose of this thesis is to study the asymptotic properties for the first and second expectation numbers of large cyclic groups. The first chapter introduces the kth expectation number. This formula allows us to determine the expected size of any group. Explicit examples and computations of the first and second expectation number are given in the second chapter. Here we show example of both cyclic and dihedral groups. In chapter three we discuss arithmetic functions which are crucial to computing the first and second expectation numbers. The fourth chapter is where we introduce and prove asymptotic results for the first expectation number of large cyclic groups. The asymptotic results for the second expectation number of cyclic groups is given in the fifth chapter. Finally, the results are summarized and future work for expectation numbers is discussed.
","abstract_html":"<p>When choosing k random elements from a group the kth expectation number is the expected size of the subgroup generated by those specific elements. The main purpose of this thesis is to study the asymptotic properties for the first and second expectation numbers of large cyclic groups. The first chapter introduces the kth expectation number. This formula allows us to determine the expected size of any group. Explicit examples and computations of the first and second expectation number are given in the second chapter. Here we show example of both cyclic and dihedral groups. In chapter three we discuss arithmetic functions which are crucial to computing the first and second expectation numbers. The fourth chapter is where we introduce and prove asymptotic results for the first expectation number of large cyclic groups. The asymptotic results for the second expectation number of cyclic groups is given in the fifth chapter. Finally, the results are summarized and future work for expectation numbers is discussed.</p>","abstract_has_math":false,"creators":["El-Farrah, Miriam Mahannah"],"institution":null,"degree_name":"Master of Science","degree_level":null,"degree_discipline":"Department of Mathematics","degree_department":null,"school":null,"contributors":["Dominic Lanphier (Director), Melanie Autin, and Molly Dunkum"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-07-01T07:00:00Z","date_published":"2015-07-01T07:00:00Z","updated_at":"2026-07-24T06:08:52Z","subjects":["Cyclic Groups","Arithmetic functions","Reimann zeta function","Algebra","Discrete Mathematics and Combinatorics","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.wku.edu/theses/1518","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dominic Lanphier (Director), Melanie Autin, and Molly Dunkum"]},{"key":"dc:creator","label":"Author","values":["El-Farrah, Miriam Mahannah"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Department of Mathematics"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Cyclic Groups","Arithmetic functions","Reimann zeta function","Algebra","Discrete Mathematics and Combinatorics","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.wku.edu/theses/1518"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["When choosing k random elements from a group the kth expectation number is the expected size of the subgroup generated by those specific elements. The main purpose of this thesis is to study the asymptotic properties for the first and second expectation numbers of large cyclic groups. The first chapter introduces the kth expectation number. This formula allows us to determine the expected size of any group. Explicit examples and computations of the first and second expectation number are given in the second chapter. Here we show example of both cyclic and dihedral groups. In chapter three we discuss arithmetic functions which are crucial to computing the first and second expectation numbers. The fourth chapter is where we introduce and prove asymptotic results for the first expectation number of large cyclic groups. The asymptotic results for the second expectation number of cyclic groups is given in the fifth chapter. Finally, the results are summarized and future work for expectation numbers is discussed.
"]},{"key":"dc:title","label":"Title","values":["Expectation Numbers of Cyclic Groups"]}]}],"canonical_facts":{"dc:contributor":["Dominic Lanphier (Director), Melanie Autin, and Molly Dunkum"],"dc:creator":["El-Farrah, Miriam Mahannah"],"dc:description.abstract":["When choosing k random elements from a group the kth expectation number is the expected size of the subgroup generated by those specific elements. The main purpose of this thesis is to study the asymptotic properties for the first and second expectation numbers of large cyclic groups. The first chapter introduces the kth expectation number. This formula allows us to determine the expected size of any group. Explicit examples and computations of the first and second expectation number are given in the second chapter. Here we show example of both cyclic and dihedral groups. In chapter three we discuss arithmetic functions which are crucial to computing the first and second expectation numbers. The fourth chapter is where we introduce and prove asymptotic results for the first expectation number of large cyclic groups. The asymptotic results for the second expectation number of cyclic groups is given in the fifth chapter. Finally, the results are summarized and future work for expectation numbers is discussed.
"],"dc:identifier":["https://digitalcommons.wku.edu/theses/1518"],"dc:subject":["Cyclic Groups","Arithmetic functions","Reimann zeta function","Algebra","Discrete Mathematics and Combinatorics","Mathematics"],"dc:title":["Expectation Numbers of Cyclic Groups"],"dc:type":["Thesis"],"thesis:degree_discipline":["Department of Mathematics"],"thesis:degree_name":["Master of Science"]},"updated_at":"2026-07-24T06:08:52Z"}