{"id":257,"date":"2026-08-29T19:16:20","date_gmt":"2026-08-29T19:16:20","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=257"},"modified":"2026-08-29T19:16:20","modified_gmt":"2026-08-29T19:16:20","slug":"group-referencing-becomes-tricky-when-large","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/08\/29\/group-referencing-becomes-tricky-when-large\/","title":{"rendered":"Group referencing becomes tricky when large"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Providing P is prime, we can refer to a [multiplicative] subgroup easily and in keeping with convention.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Let G be the multiplicative group modulo 390001<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let S be the [multiplicative] subgroup generated by 5.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">S has 24 elements as shown next<br>{ 5, 25, 125, 625, 3125, 15625, 78125, 624,<br>3120, 15600, 78000, -1, 389996, 389976, 389876, 389376,<br>386876, 374376, 311876, 389377, 386881, 374401, 312001, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">There are two problems potentially as we look at larger \/ other examples<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>The first part G can involve some very large numbers \u2013 what happens when you reach 12 digits or more?<\/li>\n\n\n\n<li>The second part where we talk of subgroups* might be problematic when the modulo is not prime.<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">*When talking of multiplicative subgroups we are usually doing this in the context of a multiplicative main group.<br>However when the modulo is composite, there is no group under multiplication from which to subgroup from.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using sturdy element notation we might refer to the above group instead as<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"610\" height=\"64\" src=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/sturdyelementnotation5forMinus5times5toTfor4.png\" alt=\"sturdy element notation for element 5 for set 1+(-1+5^t)*(5^t) for t=4\" class=\"wp-image-258\" srcset=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/sturdyelementnotation5forMinus5times5toTfor4.png 610w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/sturdyelementnotation5forMinus5times5toTfor4-300x31.png 300w\" sizes=\"auto, (max-width: 610px) 100vw, 610px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">It really is a matter of preference as to which you consider easier.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For this next 24 element example, we are hitting problem 1 (getting large) and problem 2 (composite N) so will not use a G and S way of navigating.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The 24 element multiplicative group<br>   { 5, 25, 125, 625, 3125,<br>   15625, 78125, 390625, 1953125, 9765625,<br>   48828125, 244140625, 1220703125, 6103515625,<br>   30517578125, 152587890625, 152586328121, 152578515601,<br>   152539453001, 152344140001, 151367575001,<br>   146484750001, 122070625001, 1 }<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">can only properly be described (in my opinion) using sturdy element notation as shown next.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"590\" height=\"57\" src=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/sturdyelementnotation5forPlus5times5toTfor8.png\" alt=\"sturdy element notation for element 5 for set 1+(1+5^t)*(5^t) for t=8\" class=\"wp-image-259\" srcset=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/sturdyelementnotation5forPlus5times5toTfor8.png 590w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/sturdyelementnotation5forPlus5times5toTfor8-300x29.png 300w\" sizes=\"auto, (max-width: 590px) 100vw, 590px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Where we to try to talk in modulo terms we would be saying modulo 152588281251 for any attempt to describe a main group G.<br>But we do not have a multiplicative group G?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Providing P is prime, we can refer to a [multiplicative] subgroup easily and in keeping with convention. Let G be the multiplicative group modulo 390001 Let S be the [multiplicative] subgroup generated by 5. S has 24 elements as shown next{ 5, 25, 125, 625, 3125, 15625, 78125, 624,3120, 15600, 78000, -1, 389996, 389976, 389876, [&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":[7,8],"class_list":["post-257","post","type-post","status-publish","format-standard","hentry","category-abstractalgebra","category-mathematics","tag-cyclic","tag-group"],"_links":{"self":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/257","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=257"}],"version-history":[{"count":1,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/257\/revisions"}],"predecessor-version":[{"id":260,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/257\/revisions\/260"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=257"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=257"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=257"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}