{"id":2932,"date":"2025-06-07T09:21:04","date_gmt":"2025-06-07T09:21:04","guid":{"rendered":"https:\/\/diznr.com\/?p=2932"},"modified":"2025-06-07T09:21:04","modified_gmt":"2025-06-07T09:21:04","slug":"day-06part-01-discrete-mathematics-for-gate-hindi-group-theory-and-application-its","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/day-06part-01-discrete-mathematics-for-gate-hindi-group-theory-and-application-its\/","title":{"rendered":"Day 06Part 01- Discrete Mathematics for Gate Hindi – Group theory and it’s application."},"content":{"rendered":"

Day 06Part 01- Discrete Mathematics for Gate Hindi – Group theory and it’s application.<\/p>\n

[fvplayer id=”177″]<\/p>\n

Would you like a summary of Group Theory and its Applications<\/strong> in Discrete Mathematics<\/strong> for GATE, explained in Hindi? Or are you looking for study materials, video links, or solved problems? Let me know how I can help!<\/p>\n

Here is a clear and concise guide for:<\/p>\n

\n

\ud83d\udcd8 Day 06 Part 01 \u2013 Discrete Mathematics for GATE (in Hindi)<\/strong><\/h2>\n

\ud83d\udd37 Topic: Group Theory & Its Applications<\/strong><\/h3>\n

Language<\/strong>: Hindi-English Mix (for better understanding)
Useful for<\/strong>: GATE, UGC NET, B.Tech (CSE\/IT), MCA, BCA, Competitive Exams<\/p>\n

\n

\ud83d\udd39 1. Group Theory \u0915\u094d\u092f\u093e \u0939\u0948?<\/strong><\/h2>\n

Group Theory<\/strong> \u090f\u0915 \u0910\u0938\u093e mathematical concept \u0939\u0948, \u091c\u093f\u0938\u092e\u0947\u0902 \u0939\u092e \u0915\u093f\u0938\u0940 set \u0915\u0947 elements \u092a\u0930 \u090f\u0915 operation (\u091c\u0948\u0938\u0947 addition, multiplication) apply \u0915\u0930\u0924\u0947 \u0939\u0948\u0902, \u0914\u0930 check \u0915\u0930\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u0935\u0947 \u0915\u0941\u091b rules (axioms) satisfy \u0915\u0930\u0924\u0947 \u0939\u0948\u0902 \u092f\u093e \u0928\u0939\u0940\u0902\u0964<\/p>\n

\n

\ud83d\udd38 2. Group \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e (Definition of Group)<\/strong><\/h2>\n

\u0915\u093f\u0938\u0940 non-empty set GG<\/span>G<\/span><\/span><\/span><\/span>, \u0914\u0930 \u090f\u0915 binary operation \u2217*<\/span>\u2217<\/span><\/span><\/span><\/span> \u092a\u0930 defined \u0939\u0948 \u090f\u0915 group<\/strong>, \u092f\u0926\u093f \u0935\u0939 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u091a\u093e\u0930 conditions (axioms) satisfy \u0915\u0930\u0924\u093e \u0939\u0948:<\/p>\n

\u2705 Group Axioms:<\/h3>\n
    \n
  1. \n

    Closure<\/strong>: \u2200a,b\u2208G\u21d2a\u2217b\u2208G\\forall a, b \\in G \\Rightarrow a * b \\in G<\/span>\u2200<\/span>a<\/span>,<\/span>b<\/span>\u2208<\/span><\/span>G<\/span>\u21d2<\/span><\/span>a<\/span>\u2217<\/span><\/span>b<\/span>\u2208<\/span><\/span>G<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n

  2. \n

    Associativity<\/strong>: \u2200a,b,c\u2208G\u21d2(a\u2217b)\u2217c=a\u2217(b\u2217c)\\forall a, b, c \\in G \\Rightarrow (a * b) * c = a * (b * c)<\/span>\u2200<\/span>a<\/span>,<\/span>b<\/span>,<\/span>c<\/span>\u2208<\/span><\/span>G<\/span>\u21d2<\/span><\/span>(<\/span>a<\/span>\u2217<\/span><\/span>b<\/span>)<\/span>\u2217<\/span><\/span>c<\/span>=<\/span><\/span>a<\/span>\u2217<\/span><\/span>(<\/span>b<\/span>\u2217<\/span><\/span>c<\/span>)<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n

  3. \n

    Identity Element<\/strong>: \u2203e\u2208G\\exists e \\in G<\/span>\u2203<\/span>e<\/span>\u2208<\/span><\/span>G<\/span><\/span><\/span><\/span> such that a\u2217e=e\u2217a=aa * e = e * a = a<\/span>a<\/span>\u2217<\/span><\/span>e<\/span>=<\/span><\/span>e<\/span>\u2217<\/span><\/span>a<\/span>=<\/span><\/span>a<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n

  4. \n

    Inverse Element<\/strong>: \u2200a\u2208G\\forall a \\in G<\/span>\u2200<\/span>a<\/span>\u2208<\/span><\/span>G<\/span><\/span><\/span><\/span>, \u2203a\u22121\u2208G\\exists a^{-1} \\in G<\/span>\u2203<\/span>a<\/span>\u22121<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2208<\/span><\/span>G<\/span><\/span><\/span><\/span> such that a\u2217a\u22121=a\u22121\u2217a=ea * a^{-1} = a^{-1} * a = e<\/span>a<\/span>\u2217<\/span><\/span>a<\/span>\u22121<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>a<\/span>\u22121<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2217<\/span><\/span>a<\/span>=<\/span><\/span>e<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ol>\n

    \n

    \ud83d\udca1 \u0905\u0917\u0930 \u092f\u0947 \u091a\u093e\u0930\u094b\u0902 properties satisfy \u0939\u094b\u0924\u0940 \u0939\u0948\u0902, \u0924\u094b set GG<\/span>G<\/span><\/span><\/span><\/span> \u090f\u0915 group<\/strong> \u0915\u0939\u0932\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<\/blockquote>\n

    \n

    \ud83d\udd39 3. Types of Groups<\/strong><\/h2>\n
    \n
    \n\n\n\n\n\n\n\n\n
    Type<\/th>\nCondition<\/th>\n<\/tr>\n<\/thead>\n
    Abelian Group<\/strong><\/td>\n\u092f\u0926\u093f group \u092e\u0947\u0902 commutativity \u0939\u094b i.e., a\u2217b=b\u2217aa * b = b * a<\/span>a<\/span>\u2217<\/span><\/span>b<\/span>=<\/span><\/span>b<\/span>\u2217<\/span><\/span>a<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
    Non-Abelian<\/strong><\/td>\n\u092f\u0926\u093f a\u2217b\u2260b\u2217aa * b \\ne b * a<\/span>a<\/span>\u2217<\/span><\/span>b<\/span>\ue020<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>b<\/span>\u2217<\/span><\/span>a<\/span><\/span><\/span><\/span> for some a,b\u2208Ga, b \\in G<\/span>a<\/span>,<\/span>b<\/span>\u2208<\/span><\/span>G<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
    Finite Group<\/strong><\/td>\nGroup with finite number of elements<\/td>\n<\/tr>\n
    Infinite Group<\/strong><\/td>\nGroup with infinite elements<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
    \n
    <\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n

    \ud83d\udd38 4. Example (Addition on Integers)<\/strong><\/h2>\n

    Let G=ZG = \\mathbb{Z}<\/span>G<\/span>=<\/span><\/span>Z<\/span><\/span><\/span><\/span> (integers), operation: ++<\/span>+<\/span><\/span><\/span><\/span><\/p>\n

      \n
    • \n

      Closure: a+b\u2208Za + b \\in \\mathbb{Z}<\/span>a<\/span>+<\/span><\/span>b<\/span>\u2208<\/span><\/span>Z<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n

    • \n

      Associativity: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c)<\/span>(<\/span>a<\/span>+<\/span><\/span>b<\/span>)<\/span>+<\/span><\/span>c<\/span>=<\/span><\/span>a<\/span>+<\/span><\/span>(<\/span>b<\/span>+<\/span><\/span>c<\/span>)<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n

    • \n

      Identity: 00<\/span>0<\/span><\/span><\/span><\/span>, since a+0=aa + 0 = a<\/span>a<\/span>+<\/span><\/span>0<\/span>=<\/span><\/span>a<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n

    • \n

      Inverse: \u2212a-a<\/span>\u2212<\/span>a<\/span><\/span><\/span><\/span>, since a+(\u2212a)=0a + (-a) = 0<\/span>a<\/span>+<\/span><\/span>(<\/span>\u2212<\/span>a<\/span>)<\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n

      \u2705 So, (Z,+)(\\mathbb{Z}, +)<\/span>(<\/span>Z<\/span>,<\/span>+<\/span>)<\/span><\/span><\/span><\/span> is a group
      \u2705 It is also Abelian<\/strong><\/p>\n

      \n

      \ud83d\udd37 5. Applications of Group Theory<\/strong><\/h2>\n
      \n
      \n\n\n\n\n\n\n\n\n\n
      Area<\/th>\nApplication Example<\/th>\n<\/tr>\n<\/thead>\n
      Cryptography<\/strong><\/td>\nRSA, ECC \u2013 modular arithmetic with groups<\/td>\n<\/tr>\n
      Coding Theory<\/strong><\/td>\nError detection & correction<\/td>\n<\/tr>\n
      Computer Graphics<\/strong><\/td>\nSymmetry and rotation operations<\/td>\n<\/tr>\n
      Automata Theory<\/strong><\/td>\nTransition systems and algebraic structures<\/td>\n<\/tr>\n
      Compiler Design<\/strong><\/td>\nGrammar transformations and syntax trees<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
      \n
      <\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n

      \ud83e\udde0 Important for GATE<\/strong>:<\/h2>\n
        \n
      • \n

        Order of a group (number of elements)<\/p>\n<\/li>\n

      • \n

        Subgroups<\/p>\n<\/li>\n

      • \n

        Lagrange\u2019s Theorem<\/p>\n<\/li>\n

      • \n

        Cyclic Groups<\/p>\n<\/li>\n

      • \n

        Permutation Groups<\/p>\n<\/li>\n

      • \n

        Group Tables<\/p>\n<\/li>\n

      • \n

        Application in number theory (modular groups)<\/p>\n<\/li>\n<\/ul>\n

        \n

        \u2705 Summary Table:<\/h2>\n
        \n
        \n\n\n\n\n\n\n\n\n
        Property<\/th>\nMeaning<\/th>\n<\/tr>\n<\/thead>\n
        Group<\/td>\nClosure, Associative, Identity, Inverse<\/td>\n<\/tr>\n
        Abelian<\/td>\nCommutative Group<\/td>\n<\/tr>\n
        Finite Group<\/td>\nFixed number of elements<\/td>\n<\/tr>\n
        Applications<\/td>\nCryptography, Algebra, CS Theory, Graphics<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
        \n
        <\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n

        Would you like:<\/p>\n

          \n
        • \n

          \ud83d\udcc4 GATE-style MCQs with solutions?<\/p>\n<\/li>\n

        • \n

          \ud83d\udcca Group table examples?<\/p>\n<\/li>\n

        • \n

          \ud83c\udfa5 A short explainer video in Hindi?<\/p>\n<\/li>\n<\/ul>\n

          Let me know!<\/p>\n","protected":false},"excerpt":{"rendered":"

          Day 06Part 01- Discrete Mathematics for Gate Hindi – Group theory and it’s application. [fvplayer id=”177″] Would you like a summary of Group Theory and its Applications in Discrete Mathematics for GATE, explained in Hindi? Or are you looking for study materials, video links, or solved problems? Let me know how I can help! Here […]<\/p>\n","protected":false},"author":71,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[76],"tags":[],"class_list":["post-2932","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2932","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/users\/71"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/comments?post=2932"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2932\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=2932"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=2932"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=2932"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}