{"id":165,"date":"2007-08-14T15:42:58","date_gmt":"2007-08-14T15:42:58","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=165"},"modified":"2020-03-14T19:29:30","modified_gmt":"2020-03-14T19:29:30","slug":"diagonal-stripes-in-group-table","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=165","title":{"rendered":"Diagonal Stripes in Group Table"},"content":{"rendered":"

One of my student’s attention was drawn to the fact, that in cyclic groups of order 4 and 5, it is possible to arrange the elements so that the transverse diagonals (that is those perpendicular to the leading diagonal) of the group (Cayley) table consist of equal elements. The groups in question were \\mathbb{Z}^*_5 with multiplication modulo 5 and \\mathbb{Z}_5 with addition modulo 5. Thus you get the following diagonal patterns highlighted by the colours:<\/p>\n

\\begin{array}{c|cccc} \\times_5 & 1 & 2 & 4 & 3 \\\\ \\hline 1 & 1 & \\color{green}2 & \\color{red}4 & \\color{blue}3\\\\ 2 & \\color{green}2 & \\color{red}4 & \\color{blue}3 & 1\\\\ 4 & \\color{red}4 &\\color{blue}3 & 1 & \\color{green}2\\\\ 3 & \\color{blue}3 & 1 & \\color{green}2 & \\color{red}4\\\\ \\end{array} \\qquad\\qquad \\begin{array}{c|cccccc} +_5 & 0 & 1& 2 & 3 & 4 \\\\ \\hline 0 & 0 & \\color{green}1 & \\color{red}2 & \\color{blue}3 & \\color{magenta}4 \\\\ 1 & \\color{green}1 & \\color{red}2 & \\color{blue}3 & \\color{magenta}4 & 0 \\\\ 2 & \\color{red}2 &\\color{blue}3 & \\color{magenta}4 & 0 & \\color{green}1\\\\ 3 & \\color{blue}3 & \\color{magenta}4 & 0 & \\color{green}1 & \\color{red}2\\\\ 4 & \\color{magenta}4 & 0 & \\color{green}1 & \\color{red}2 & \\color{blue}3 \\end{array}<\/p>\n

He asked 2 questions:<\/p>\n

1. Can the elements of all (finite) cyclic groups be arranged to give these diagonal stripes?<\/em><\/p><\/blockquote>\n

If you take a finite cyclic group generated by the element a (we will use multiplication for the binary operation) then the natural ordering e, a ,a^2 , a^3, \\dots, a^n will show this pattern:<\/p>\n

\\begin{array}{c|llllllll} & e & a & a^2 & & & a^{i-1} & a^i & \\\\ \\hline e & e & \\color{green}a & \\color{red}a^2 & & & & \\color{blue}a^i \\\\ a & \\color{green}a & \\color{red}a^2 & & & & \\color{blue}a^i \\\\ a^2 & \\color{red}a^2 & & & &\\color{blue}a^i\\\\ \\\\ a^{i-1} & & \\color{blue}a^i\\\\ a^i & \\color{blue}a^i\\\\ \\\\ a^{j-1} & & & & & & & \\color{magenta}a^{j-1+i}\\\\ a^j & & & & & & \\color{magenta}a^{j+i-1}\\\\ \\ \\end{array}<\/p>\n

2. Are the cyclic groups the only ones that generate these patterns?<\/em><\/p><\/blockquote>\n

The answer is yes, but is not quite so obvious although not difficult to prove. Suppose a (finite) group G arranged as G=\\{e,a_1,a_2,\\dots,a_n\\} exhibits the diagonal stripes. Use induction. Let a_1=a and suppose also that a_j=a^j for 1 \\le j&lt;i. Then we get:
\n\\begin{array}{c|lllllll} & e & a & & & a_{i-1} & a_i & \\\\ \\hline e & & & & & & \\color{blue}a_i & \\\\ a & & & & & \\color{blue}aa_{i-1} & \\\\ \\ \\end{array}
\nIt follows that \\color{blue}a_i=aa_{i-1} so by induction assumption, a_i=aa^{i-1}=a^i and hence G is the cyclic group generated by a.<\/p>\n","protected":false},"excerpt":{"rendered":"

One of my student’s attention was drawn to the fact, that in cyclic groups of order 4 and 5, it is possible to arrange the elements so that the transverse diagonals (that is those perpendicular to the leading diagonal) of the group (Cayley) table consist of equal elements. The groups in question were with multiplication […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-165","post","type-post","status-publish","format-standard","hentry","category-articles"],"_links":{"self":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/165","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=165"}],"version-history":[{"count":1,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/165\/revisions"}],"predecessor-version":[{"id":281,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/165\/revisions\/281"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=165"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=165"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=165"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}