{"id":299,"date":"2026-09-03T18:29:19","date_gmt":"2026-09-03T18:29:19","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=299"},"modified":"2026-09-03T18:29:19","modified_gmt":"2026-09-03T18:29:19","slug":"sure-its-structure-preserving","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/09\/03\/sure-its-structure-preserving\/","title":{"rendered":"Sure it\u2019s structure preserving"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">An isomorphism is said to be structure preserving<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>In fact relationships between elements are maintained even if the elements are renamed or reordered<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>But what if a finite multiplicative cyclic group has a secondary feature?<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Allowing both relabelling and reordering means this secondary feature is no longer accessible or usable.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Does this prevent the mapping being structure preserving?<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A truly structure preserving mapping would allow the use of the secondary feature even after reordering and relabelling.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Best illustrated with examples.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using a generating element of 65 and working modulo 8321 we obtain the 52 element multiplicative group shown next.<br>   { 65,4225,32,<strong>2080<\/strong>,2064,1024,8313,7801,7805,8065,<br>   2,<strong>130<\/strong>,129,64,4160,4128,2048,8305,7281,<strong>7289<\/strong>,<br>   7809,4,260,258,128,8320,8256,<strong>4096<\/strong>,8289,6241,<br>   6257,7297,8,520,516,<strong>256<\/strong>,8319,8191,8192,8257,<br>   4161,4193,6273,<strong>16<\/strong>,1040,1032,512,<strong>8317<\/strong>,8061,8063,<br>   8193,1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">There is a secondary feature in that group that allows us to jump to a portion of our group and be at most 7 elements from the element we require.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Every eighth element after 16 from the end of the group is a power of 2.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In order we would write them as follows<br><\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msup><mn>2<\/mn><mn>4<\/mn><\/msup><mo separator=\"true\">,<\/mo><msup><mn>2<\/mn><mn>8<\/mn><\/msup><mo separator=\"true\">,<\/mo><msup><mn>2<\/mn><mn>12<\/mn><\/msup><mo separator=\"true\">,<\/mo><msup><mn>2<\/mn><mn>16<\/mn><\/msup><mo separator=\"true\">,<\/mo><msup><mn>2<\/mn><mn>20<\/mn><\/msup><mo separator=\"true\">,<\/mo><msup><mn>2<\/mn><mn>24<\/mn><\/msup><mo separator=\"true\">,<\/mo><msup><mn>2<\/mn><mn>28<\/mn><\/msup><mo separator=\"true\">,<\/mo><msup><mn>2<\/mn><mn>32<\/mn><\/msup><mo separator=\"true\">,<\/mo><msup><mn>2<\/mn><mn>36<\/mn><\/msup><mo separator=\"true\">,<\/mo><msup><mn>2<\/mn><mn>40<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">2^4, 2^8, 2^{12}, 2^{16}, 2^{20}, 2^{24}, 2^{28}, 2^{32}, 2^{36}, 2^{40}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Writing them longhand we might say<br>   eight from last is 16<br>   sixteenth from last is 2^8<br>   twentyfourth from last is 2^12<br>   thirtysecond from last is 2^16<br>   fortieth from last is 2^20<br>   fortyeighth from last is 2^24<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The fourth item is in bold in the group listing above and represents 2^24 mod 8321<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using sturdy element notation we might say the group we tabulated above beginning 65 is<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"711\" height=\"75\" src=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1plus2forproth6.png\" alt=\"Sturdy element notation for 1+2^t for set 1+(1+2^t)*(2^(t+1)) for t=6\" class=\"wp-image-300\" srcset=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1plus2forproth6.png 711w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1plus2forproth6-300x32.png 300w\" sizes=\"auto, (max-width: 711px) 100vw, 711px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Current theory tells us that there exists an isomorphism between this finite abelian cyclic group and the additive group Z52.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Such an isomorphism would use both relabelling and reordering.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The problem then becomes how to access the secondary feature?<br>We are now [after isomorphism] the group Z52 whose elements are:<br>   { 1,2,3,4,5,6,7,8,9,10,11,12,13,<br>   14,15,16,17,18,19,20,21,22,23,24,25,26,<br>   27,28,29,30,31,32,33,34,35,36,37,38,39,<br>   40,41,42,43,44,45,46,47,48,49,50,51,52 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By allowing reordering without being specific about tracking each element and how it maps individually, we have no way of locating an element of the secondary feature once in Z52<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>This prevents us from jumping to a section of the group and navigating a maximum of seven elements from that jump destination.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">That secondary feature is part of our original multiplicative cyclic group.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We are unable to use that secondary feature.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Therefore we are not truly structure preserving.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Next a further example.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using a generating element of 257 and working modulo 131585 we obtain the 68 element multiplicative group shown next.<br>   { 257, 66049, 128, <strong>32896<\/strong>, 32832, 16384, 131553, 123361,<br>   123377, 127489, 8, <strong>2056<\/strong>, 2052, 1024, 131583, 131071,<br>   131072, 131329, 65793, <strong>65921<\/strong>, 98817, 64, 16448, 16416,<br>   8192, 131569, 127473, <strong>127481<\/strong>, 129537, 4, 1028, 1026, <br>   512, <strong>-1<\/strong>, 131328, <strong>65536<\/strong>, 131457, 98689, 98753, 115201,<br>   32, 8224, 8208, <strong>4096<\/strong>, 131577, 129529, 129533, 130561,<br>   2, 514, 513, <strong>256<\/strong>, 65792, 65664, 32768, 131521,<br>   115137, 115169, 123393, <strong>16<\/strong>, 4112, 4104, 2048, <strong>131581<\/strong>,<br>   130557, 130559, 131073, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using sturdy element notation we might say the group we tabulated above beginning 257 is<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"709\" height=\"67\" src=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1plus2prothYfor8.png\" alt=\"Sturdy element notation for 1+2^t for set 1+(1+2^t)*(2^(t+1)) for t=8\" class=\"wp-image-301\" srcset=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1plus2prothYfor8.png 709w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/sturdyelementnotation1plus2prothYfor8-300x28.png 300w\" sizes=\"auto, (max-width: 709px) 100vw, 709px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Labelling that 68 element multiplicative cyclic group as TPy68.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>With a jump or secondary feature similar to our previous example<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">   eight from last is 16<br>   sixteenth from last is 2^8<br>   twentyfourth from last is 2^12<br>   thirtysecond from last is 2^16<br>   fortieth from last is 2^20<br>   fortyeighth from last is 2^24<br>   fiftysixth from last is 2^28<br>   sixtyfourth from last is 2^32<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The fourth item is in bold in the group listing above and represents 2^32 mod 131585.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Current theory tells us that there exists an isomorphism between this finite abelian cyclic group and the additive group Z68.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Such an isomorphism would use both relabelling and reordering.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The problem then becomes how to access the secondary feature?<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We are now [after isomorphism] the group Z68 whose elements are<br>   { 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,<br>   18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,<br>   35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,<br>   52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By allowing reordering without being specific about tracking each element and how it maps individually, we have no way of locating an element of the secondary feature once in Z68<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>This prevents us from jumping to a section of the group and navigating a maximum of seven elements from that jump destination.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">That secondary feature is part of our original multiplicative cyclic group TPy68.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We are unable to use that secondary feature.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore we are not truly structure preserving.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If the jump feature is an essential part of the multiplication operation, then you cannot have a structure preserving map.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Do we make a special case for this type of group, or do we modify or qualify current theory?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>A question that might be hard to answer is do we in fact have the jump feature in the Z group and just need to add something to the labelling to allow it to operate [again]?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Want to try and highlight in bold the elements in the Z group? <br>Would that be enough?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>An isomorphism is said to be structure preserving In fact relationships between elements are maintained even if the elements are renamed or reordered But what if a finite multiplicative cyclic group has a secondary feature? Allowing both relabelling and reordering means this secondary feature is no longer accessible or usable. Does this prevent the mapping [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12,5],"tags":[7,11,8],"class_list":["post-299","post","type-post","status-publish","format-standard","hentry","category-abstractalgebra","category-mathematics","tag-cyclic","tag-definition","tag-group"],"_links":{"self":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/299","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/comments?post=299"}],"version-history":[{"count":1,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/299\/revisions"}],"predecessor-version":[{"id":302,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/299\/revisions\/302"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=299"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=299"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=299"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}