%0 Journal Article
%T Average Probability of an Element Being a Generator in the Cyclic Group
%A Yoshihiro Tanaka
%J American Journal of Computational Mathematics
%P 230-235
%@ 2161-1211
%D 2023
%I Scientific Research Publishing
%R 10.4236/ajcm.2023.132012
%X All
elements in the cyclic group&nbsp;<img src=\"Edit_61cff16c-1809-4919-b9f1-a32efccf5fcc.png\" alt=\"\" />&nbsp;<span></span><span><span><span style=\"font-family:;\" \"=\"\"><span> </span>are generated
by a generator </span></span></span><span><span><i><span style=\"font-family:;\" \"=\"\">g</span></i></span></span><span><span><span style=\"font-family:;\" \"=\"\">. The
number of generators of&nbsp;<img src=\"Edit_a581344a-bdfa-43fc-93c0-34e7ed0b6a62.png\" alt=\"\" />&nbsp;</span><span></span></span></span><span><span><span style=\"font-family:;\" \"=\"\"><span></span>of&nbsp;<img src=\"Edit_10696e74-29a0-4b94-843c-96c9f21bfe07.png\" alt=\"\" />&nbsp;</span><span></span></span></span><span><span><span style=\"font-family:;\" \"=\"\">, namely&nbsp;<img src=\"Edit_d25b0a87-55c0-4b42-84c2-a92d936e81ec.png\" alt=\"\" />&nbsp;</span><span></span></span></span><span><span><span style=\"font-family:;\" \"=\"\"><span> </span>is known to
be Euler¡¯s totient function <img src=\"Edit_90000458-2800-4133-be87-27cc248944b3.png\" alt=\"\" /></span><span></span></span></span><span><span><span style=\"font-family:;\" \"=\"\">; however, the average probability of an element
being a generator has not been discussed before. Several
analytic properties of&nbsp;<img src=\"Edit_b10544dc-13aa-4841-b3db-ee9f8cea7bc5.png\" alt=\"\" />&nbsp;</span><span><span></span></span></span></span><span><span><span><span style=\"font-family:;\" \"=\"\"><span> </span>have been investigated for a long time.
However, it seems that some issues still remain unresolved.</span></span></span></span><span><span><span style=\"font-family:;\" \"=\"\"> In this
study, we derive the average probability of an element being a generator using previous classical studies.</span></span></span>
%K Generator
%K Cyclic Group
%K Euler Product
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=125503