{"id":4486,"date":"2025-08-30T12:53:58","date_gmt":"2025-08-30T12:53:58","guid":{"rendered":"https:\/\/journal.ziu-university.net\/?p=4486"},"modified":"2025-08-30T12:53:59","modified_gmt":"2025-08-30T12:53:59","slug":"from-structure-to-application-employing-abelian-groups-in-contemporary-number-theory","status":"publish","type":"post","link":"https:\/\/journal.ziu.edu.sy\/?p=4486","title":{"rendered":"From Structure to Application \u00a0Employing Abelian Groups in contemporary Number theory"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><strong>Abstract:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This research investigates the algebraic structure of Abelian groups by exploring their core properties and practical applications in number theory. The study emphasizes that the commutative nature of these groups is not only a theoretical interest but a foundational tool for solving various number-theoretic problems, such as Diophantine equations, congruences, and prime analysis, as well as in modern applications including cryptography and coding theory. Through analytical exploration and illustrative examples, the study demonstrates that Abelian groups provide a flexible and powerful mathematical framework suitable for constructing effective, real-world models. The findings confirm that integrating algebraic theory with applied contexts enhances mathematical understanding and supports the development of high-efficiency mathematical and security solutions.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Keywords: Abelian Groups, Abstract Algebra, Number Theory, Cryptography, Diophantine Equations, Coding Theory.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u0627\u0644\u0645\u0644\u062e\u0635<\/strong>:&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u064a\u062a\u0646\u0627\u0648\u0644 \u0647\u0630\u0627 \u0627\u0644\u0628\u062d\u062b \u062f\u0631\u0627\u0633\u0629 \u0627\u0644\u0628\u0646\u064a\u0629 \u0627\u0644\u062c\u0628\u0631\u064a\u0629 \u0644\u0644\u0632\u0645\u0631 \u0627\u0644\u062a\u0628\u0627\u062f\u0644\u064a\u0629 (Abelian Groups) \u0645\u0646 \u062e\u0644\u0627\u0644 \u0627\u0633\u062a\u0643\u0634\u0627\u0641 \u062e\u0648\u0627\u0635\u0647\u0627 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u0648\u062a\u0637\u0628\u064a\u0642\u0627\u062a\u0647\u0627 \u0627\u0644\u0639\u0645\u0644\u064a\u0629 \u0641\u064a \u0646\u0638\u0631\u064a\u0629 \u0627\u0644\u0623\u0639\u062f\u0627\u062f. \u0627\u0646\u0637\u0644\u0642\u062a \u0627\u0644\u062f\u0631\u0627\u0633\u0629 \u0645\u0646 \u0641\u0643\u0631\u0629 \u0623\u0646 \u0627\u0644\u0637\u0627\u0628\u0639 \u0627\u0644\u062a\u0628\u0627\u062f\u0644\u064a \u0644\u0647\u0630\u0647 \u0627\u0644\u0632\u0645\u0631 \u0644\u0627 \u064a\u0642\u062a\u0635\u0631 \u0639\u0644\u0649 \u0627\u0644\u062c\u0627\u0646\u0628 \u0627\u0644\u0646\u0638\u0631\u064a \u0641\u062d\u0633\u0628\u060c \u0628\u0644 \u064a\u0634\u0643\u0644 \u0623\u0633\u0627\u0633\u064b\u0627 \u0641\u064a \u062d\u0644 \u0639\u062f\u062f \u0645\u0646 \u0627\u0644\u0645\u0634\u0643\u0644\u0627\u062a \u0627\u0644\u0639\u062f\u062f\u064a\u0629 \u0645\u062b\u0644 \u0627\u0644\u0645\u0639\u0627\u062f\u0644\u0627\u062a \u0627\u0644\u062f\u064a\u0648\u0641\u0627\u0646\u062a\u064a\u0629\u060c \u0648\u0627\u0644\u062a\u062d\u0644\u064a\u0644 \u0627\u0644\u062a\u0648\u0627\u0641\u0642\u064a\u060c \u0648\u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0623\u0648\u0644\u064a\u0629\u060c \u0648\u0643\u0630\u0644\u0643 \u0641\u064a \u0645\u062c\u0627\u0644\u0627\u062a \u062a\u0637\u0628\u064a\u0642\u064a\u0629 \u062d\u062f\u064a\u062b\u0629 \u0643\u0627\u0644\u062a\u0634\u0641\u064a\u0631 \u0648\u0646\u0638\u0631\u064a\u0629 \u0627\u0644\u062a\u0631\u0645\u064a\u0632. \u0645\u0646 \u062e\u0644\u0627\u0644 \u062a\u062d\u0644\u064a\u0644 \u0646\u0638\u0631\u064a \u0648\u0623\u0645\u062b\u0644\u0629 \u0648\u0627\u0642\u0639\u064a\u0629\u060c \u0623\u062b\u0628\u062a\u062a \u0627\u0644\u062f\u0631\u0627\u0633\u0629 \u0623\u0646 \u0627\u0644\u0632\u0645\u0631 \u0627\u0644\u062a\u0628\u0627\u062f\u0644\u064a\u0629 \u062a\u0648\u0641\u0631 \u0625\u0637\u0627\u0631\u064b\u0627 \u0631\u064a\u0627\u0636\u064a\u064b\u0627 \u0645\u0631\u0646\u064b\u0627 \u064a\u064f\u0645\u0643\u0650\u0651\u0646 \u0645\u0646 \u0628\u0646\u0627\u0621 \u0646\u0645\u0627\u0630\u062c \u0641\u0639\u0627\u0644\u0629 \u0648\u0642\u0627\u0628\u0644\u0629 \u0644\u0644\u062a\u0637\u0628\u064a\u0642 \u0641\u064a \u0628\u064a\u0626\u0627\u062a \u0645\u062a\u0639\u062f\u062f\u0629. \u062a\u0648\u0635\u0644 \u0627\u0644\u0628\u062d\u062b \u0625\u0644\u0649 \u0623\u0646 \u062f\u0645\u062c \u0627\u0644\u0646\u0638\u0631\u064a\u0629 \u0627\u0644\u062c\u0628\u0631\u064a\u0629 \u0628\u0627\u0644\u0648\u0627\u0642\u0639 \u0627\u0644\u062a\u0637\u0628\u064a\u0642\u064a \u064a\u0639\u0632\u0632 \u0645\u0646 \u0641\u0647\u0645 \u0627\u0644\u0631\u064a\u0627\u0636\u064a\u0627\u062a \u0627\u0644\u062d\u062f\u064a\u062b\u0629 \u0648\u064a\u064f\u0633\u0647\u0645 \u0641\u064a \u062a\u0637\u0648\u064a\u0631 \u062d\u0644\u0648\u0644 \u0631\u064a\u0627\u0636\u064a\u0629 \u0648\u0623\u0645\u0646\u064a\u0629 \u0630\u0627\u062a \u0643\u0641\u0627\u0621\u0629 \u0639\u0627\u0644\u064a\u0629.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u0627\u0644\u0643\u0644\u0645\u0627\u062a \u0627\u0644\u0645\u0641\u062a\u0627\u062d\u064a\u0629: \u0627\u0644\u0632\u0645\u0631 \u0627\u0644\u062a\u0628\u0627\u062f\u0644\u064a\u0629\u060c \u0627\u0644\u062c\u0628\u0631 \u0627\u0644\u0645\u062c\u0631\u062f\u060c \u0627\u0644\u062a\u0631\u0645\u064a\u0632\u060c \u0646\u0638\u0631\u064a\u0629 \u0627\u0644\u0627\u0639\u062f\u0627\u062f\u060c \u0627\u0644\u062a\u0634\u0641\u064a\u0631 .&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">by : <\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><td><strong>Omalkhear Salem ALmabrwk Bleblou<\/strong><br><strong>\u00a0\u0627\u0644\u062f\u0643\u062a\u0648\u0631\u0629 \u0627\u0645 \u0627\u0644\u062e\u064a\u0631\u0633\u0627\u0644\u0645 \u0627\u0644\u0645\u0628\u0631\u0648\u0643 \u0628\u0644\u064a\u0628\u0644\u0648<\/strong><\/td><\/tr><tr><td>\u0627\u0644\u0642\u0633\u0645 \u0627\u0644\u0639\u0627\u0645 &#8211; \u0643\u0644\u064a\u0629 \u0627\u0644\u0639\u0644\u0648\u0645 \u0627\u0644\u0635\u062d\u064a\u0629-\u0627\u0644\u0639\u062c\u064a\u0644\u0627\u062a &#8211; \u062c\u0627\u0645\u0639\u0629 \u0627\u0644\u0632\u0627\u0648\u064a\u0629 -\u0644\u064a\u0628\u064a\u0627<\/td><\/tr><\/tbody><\/table><\/figure>\n\r\n\t\t<div class=\"review_wrap\">\r\n\t\t\t<div id=\"review-box\" class=\"review-box review-bottom review-stars\">\r\n\t\t\t\t<div class=\"review-summary\">\r\n\t\t\t\t\t<div class=\"review-final-score\">\r\n\t\t\t\t\t\t<span title=\"\" class=\"post-large-rate stars-large\"><span style=\"width:0%\"><\/span><\/span>\r\n\t\t\t\t\t\t<h4><\/h4>\r\n\t\t\t\t\t<\/div>\r\n\t\t\t\t\r\n\t\t\t\t<div class=\"review-short-summary\"><a href=\"https:\/\/drive.google.com\/file\/d\/1kdA6wRLo9O95FNZFJ3hPUMoTNbPhtAzH\/view?usp=drive_link\" class=\"taq-button taq-medium taq-square taq-flat\" style=\"background-color:#a0ce4e\" target=\"_blank\"><i class=\"fa fa-download\"><\/i><span class=\"button-text\">Download<\/span><\/a>\r\n\t\t\t\t<\/div>\r\n\t\t\t<\/div>\r\n\t\t\t\r\n\t\t\t<div class=\"user-rate-wrap\">\r\n\t\t\t\t<span class=\"user-rating-text\">\r\n\t\t\t\t\t<strong>\u062a\u0642\u064a\u064a\u0645 \u0627\u0644\u0645\u0633\u062a\u062e\u062f\u0645\u0648\u0646: <\/strong>\r\n\t\t\t\t\t<span class=\"taq-score\"><\/span>\r\n\t\t\t\t\t<small>\u0643\u0646 \u0623\u0648\u0644 \u0627\u0644\u0645\u0635\u0648\u062a\u0648\u0646 !<\/small>\r\n\t\t\t\t<\/span>\r\n\r\n\t\t\t\t<div data-rate=\"0\" data-id=\"4486\" class=\"user-rate taq-user-rate-active\">\r\n\t\t\t\t\t<span class=\"user-rate-image post-large-rate stars-large\">\r\n\t\t\t\t\t\t<span style=\"width:0%\"><\/span>\r\n\t\t\t\t\t<\/span>\r\n\t\t\t\t<\/div>\r\n\r\n\t\t\t\t<div class=\"taq-clear\"><\/div>\r\n\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t<\/div>","protected":false},"excerpt":{"rendered":"<p>Abstract: This research investigates the algebraic structure of Abelian groups by exploring their core properties and practical applications in number theory. The study emphasizes that the commutative nature of these groups is not only a theoretical interest but a foundational tool for solving various number-theoretic problems, such as Diophantine equations, congruences, and prime analysis, as &hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[147],"tags":[],"class_list":["post-4486","post","type-post","status-publish","format-standard","","category----1"],"_links":{"self":[{"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=\/wp\/v2\/posts\/4486","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4486"}],"version-history":[{"count":1,"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=\/wp\/v2\/posts\/4486\/revisions"}],"predecessor-version":[{"id":4487,"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=\/wp\/v2\/posts\/4486\/revisions\/4487"}],"wp:attachment":[{"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4486"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4486"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4486"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}