maths.free › Abstract Algebra › Cyclic groups and permutation groups

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.

Намунаи коркардашуда · inverse of 5 mod 12

Number-Theory · Modular-Arithmetic

Inverse of 5 mod 12

5,\ 12

Қадами ба қадам

  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.

Ҷавоби ҷавобро нишон диҳед
5^{-1} \equiv 5 \pmod{12}

Ҳоло шумо Масъаларо интихоб кунед, ё ки онро навишта ё кашед. Ҳар қадам, тасвир, ҷавоб то он даме, ки шумо пурсед, пинҳон мешавад.

Кор кардан

Дар ҳисоби ройгон ба ҳар як дарс қайдҳо илова карда мешаванд, қайди он, ки шумо чӣ кор кардаед, масъалаҳои ҳалшуда дар як ҷо ва муаллиме, ки шумо метавонед дар бораи ин саҳифа пурсед. Математика барои ҳама кушода аст, хоҳ ворид шуда бошад ё на.

Бақайдгирӣ Ворид шудан

Символы, используемые здесь

Барои муайянкунии пурраи маъно, тасвир ва маънои ҳар як ҳарф дар он, ба ҳар як аломат пахш кунед.

Чӣ тавр: 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.

Саволҳои маъмул

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.

& Иловаи забон Abstract Algebra

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