{"id":249,"date":"2020-04-16T10:15:55","date_gmt":"2020-04-16T10:15:55","guid":{"rendered":"http:\/\/blogs.rollins.edu\/graphembeddings\/?page_id=249"},"modified":"2020-05-05T08:37:46","modified_gmt":"2020-05-05T08:37:46","slug":"groups","status":"publish","type":"page","link":"https:\/\/blogs.rollins.edu\/graphembeddings\/embedding-complete-bipartite-graphs-with-cayley-maps\/groups\/","title":{"rendered":"Groups"},"content":{"rendered":"\n\n\n

Group <\/strong>
A group \"G\" is a set with a binary operation * , such that
1. The set has an identity element \"e\" (i.e., 

*** QuickLaTeX cannot compile formula:\ng</em>&e=e*<em>g=g\n\n*** Error message:\nMisplaced alignment tab character &.\r\nleading text: $g</em>&\r\n\n<\/pre> \"\forall\" \"g \in G\"). <\/em>
2. Each element \"g \in G\" has an inverse \"g^{-1}\" (i.e., \"g</em>*g^{-1} = g^{-1}*g = e\").<\/p>\n\n\n\n
Generating Set of a Finite Group<\/strong>
The generating set \"S\" is a set of elements in a finite group \"G\" such that every element in \"G\" is a product of the elements in \"S\". That is, the entire group can be formed using just the elements in \"S\" and the group’s operation. A generating set \"S\" is closed if for any \"s\" in \"S\", \"s^{-1}\" \"\in\" \"S\".<\/p>\n\n\n\n
\n\n\n\n
Groups for \"K_{p,p}\"<\/strong>
We focused on complete bipartite graphs with a prime number, \"p\", of vertices in each set. Cayley Graphs that represent complete bipartite graphs \"K_{p,p}\" use groups that have \"2p\" vertices. The only groups with \"2p\" vertices are \"Z_{2p}\", \"D_p\", and \"Z_2\" x \"Z_p\". We only need to consider \"Z_{2p}\" and \"D_p\" when making Cayley Maps for \"K_{p,p}\", because \"Z_2p\" and \"Z_2\" x \"Z_p\" are isomorphic, unless \"p=2\", and in that case \"D_2\" and \"Z_2\" x \"Z_2\" are isomorphic.<\/p>\n\n\n\n
Cosets<\/strong>
Within each group \"G\" there exists a subgroup, and the remaining vertices in the group form a coset, \"C\". \"C\" is equal to the closed generating set of \"G\", so its elements are used to form the rotation used in the Cayley Map \"C_M(G,\rho)\". In the context of the complete bipartite graph \"K_{p,p}\", the vertices that are in the same set as the identity element form a subgroup \"H\" of order \"p\", and the other \"p\" elements form a coset of \"H\".<\/p>\n\n\n\n
Cyclic Groups<\/strong>
Suppose \"S'\" is a closed generating set for \"Z_{2p}\", and \"C_G(Z_{2p},S')\" is a Cayley Graph for \"K_{p,p}\". The only subgroup in \"Z_{2p}\" with order \"p\" is the set of even numbers \"{0, \dots ,2p-2}\". Therefore, the coset of \"Z_{2p}\", and thus the rotation of \"C_M(Z_{2p},\rho)\", is comprised of all odd numbers \"\{1, \dots , 2p - 1\}\".<\/p>\n\n\n\n
Dihedral Group<\/strong>
Suppose \"S'\" is a closed generating set for \"D_p\", and \"C_G(D_{p},S')\" is a Cayley Graph for \"K_{p,p}\". Since the only subgroup of order \"p\" in \"D_p\" is the set of rotations, the coset of \"D_p\" is the set of reflections. Therefore, \"S' = \{ r^if : 0 \le i < p\}\". Because reflections have order 2, every element in the rotation is of order 2.<\/p>\n","protected":false},"excerpt":{"rendered":"
Group A group $G$ is a set with a binary operation * , such that 1. The set has an identity element $e$ (i.e., $g&e=e*g=g$ $\\forall$ $g \\in G$). 2. Each element $g \\in G$ has an inverse $g^{-1}$ (i.e., $g*g^{-1} = g^{-1}*g = e$). Generating Set of a Finite GroupThe generating set $S$ is […]<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":97,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-249","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/249","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/comments?post=249"}],"version-history":[{"count":18,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/249\/revisions"}],"predecessor-version":[{"id":406,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/249\/revisions\/406"}],"up":[{"embeddable":true,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/97"}],"wp:attachment":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/media?parent=249"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}