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Cyclic groups and permutation groups

Generators, orders, cycles, transpositions, parity.

A cyclic group is generated by one element; every permutation is a product of disjoint cycles, and its parity (even/odd) is well defined. Picture it: shuffling 5 cards as arrows between positions; a cycle is a loop of arrows. Think it: Cayley's theorem, every group is a permutation group, means permutations are the universal example.

उदाहरण · inverse of 5 mod 12

Number-Theory · Modular-Arithmetic

Inverse of 5 mod 12

5,\ 12

चरणद्वारा चरण

  1. 5x \equiv 1 \pmod{12}

    We want x with a·x ≡ 1 (mod m). It exists only when gcd(a, m) = 1.

  2. 12 = 2 \times 5 + 2

    Euclid step.

  3. 5 = 2 \times 2 + 1

    Euclid step.

  4. 2 = 2 \times 1 + 0

    Euclid step.

  5. 5 \times 5 = 25 \equiv 1 \pmod{12}

    Back-substitute (extended Euclid) to express 1 as a combination of a and m; the coefficient of a is the inverse.

जवाफ प्रकट गर्नुहोस्
5^{-1} \equiv 5 \pmod{12}

अब तपाईँ समस्या रोज्नुहोस्, वा टाइप गर्नुहोस् वा तपाईँको आफ्नै रेखाचित्र गर्नुहोस् । प्रत्येक चरण, एउटा तस्वीर, तपाईँले सोध्नु अघि उत्तर लुकेको छ ।

तपाईँको आफ्नै काम गर्ने राख्नुहोस्

एक निःशुल्क खाता प्रत्येक पाठ मा नोट थप्छ, तपाईं के समाप्त गरेको छ को एक रेकर्ड, एक ठाउँमा आफ्नो समाधान समस्या, र एक शिक्षक तपाईं यो पृष्ठ बारेमा सोध्न सक्नुहुन्छ. गणित आफैलाई सबैलाई खुला छ, मा साइन इन वा छैन.

साइन अप गर्नुहोस् लगइन

यहाँ प्रयोग गरिएको प्रतीक

पूर्ण परिभाषा, एउटा तस्वीर, र यो प्रत्येक अक्षर अर्थ के लागि कुनै पनि प्रतीक ट्याप गर्नुहोस्।

कसरी: Cyclic groups and permutation groups

  1. We want x with a·x ≡ 1 (mod m). It exists only when gcd(a, m) = 1.
  2. Euclid step.
  3. Euclid step.
  4. Euclid step.
  5. Back-substitute (extended Euclid) to express 1 as a combination of a and m; the coefficient of a is the inverse.

प्रश्नहरू मानिसहरूले सोध्छन्

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

यसमा थप Abstract Algebra

GroupsSubgroups, cosets and Lagrange's theoremHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula